A S KOMPAN EYETS THEORETICAL PHYSICS TRANSLATED FROM THE RUSSIAN EDITED BY OEORGE YANKOVSKY This translation has been read and approved by the author, Professor A. S. Kompaneyets Printed in the Union of Soviet Socialist Republics CONTENTS Page From the Preface to the First Edition. 7 Preface to the Second Edition. 9 Part I. Mechanics . 11 Sec. 1. Generalized Coordinates. 11 Sec. 2. Lagrange’s Equation. 13 Sec. 3. Examples of Lagrange’s Equations .24 Sec. 4. Conservation Laws .30 Sec. 6. Motion in a Central Field.41 Sec. 6. Collision of Particles.48 Sec. 7. Small Oscillations.57 Sec. 8. Rotating Coordinate Systems. Inertial Forces.66 Sec. 9. The Dynamics of a Rigid Body.73 Sec. 10. General Principles of Mechanics.81 Part 11. Electrodynamics .92 Sec. 11. Vector Analysie.92 Sec. 12. The Eleotromagnetic Field. Maxwell’s Equations.104 Sec. 13. The Action Principle for the Electromagnetic Field.117 Sec. 14. The Electrostatics of Point Charges. Slowly Varying Fields . . 124 Sec. 15. The Magnetostatics of Point Charges.135 Sec. 18. Electrodynamics of Material Media.144 Sec. 17i Plane Eleotromagnetic Waves.162 Sec. 18. Transmission of Signals. Almost Plane Waves.173 Sec. 19. The Emission of Electromagnetic Waves .181 Sec. 20. The Theory of Relativity.190 Sec. 21. Relativistic Dynamics.211 Fart III. Quantum Mechanics.229 Sec. 22. The Inadequacy of Classical Mechanics. The Analogy Between Mechanics and Geometrical Optics.229 Sec. 23. Electron Diffraction .238 Sec. 24. The Wave Equation.244 6 CONTENTS Page Sec. 25. Certain Problems of Quantum Mechanics.252 Sec. 26. Harmonic Oscillatory Motion in Quantum Mechanics (Linear Harmonic Oscillator).265 Sec. 27. Quantization of the Electromagnetic Field.271 Sec. 28. Quasi-Classical Approximation.280 Sec. 29. Operators in Quantiun Mechanics.291 Sec. 30. Expansions into Wave Functions.301 Sec. 31. Motion in a Central Field.312 Sec. 32. Electron Spin.323 Sec. 33. Many-Eleotron Systems.334 Sec. 34. The Quantum Theory of Radiation.353 Sec. 35. The Atom in a Constant External Field.368 Sec. 36. Quantum Theory of Dispersion.379 Sec. 37. Quantum Theory of Scattering.385 Sec. 38. The Relativistic Wave Equation for an Electron.394 Part IV. Statistical Physics.413 Sec. 39. The Equilibrium Distribution of Molecules in an Ideal Cas . . 413 Sec. 40. Boltzmann Statistics (Translational Motion of a Molecule. Gas in an External Field).430 Sec. 41. Boltzmann Statistics (Vibrational and Rotational Molecular Motion) .447 Sec. 42. The Application of Statistics to the Electromagnetic Field and to Crystalline Bodies.457 Sec. 43. Bose Distribution .474 Sec. 44. Fermi Distribution.477 Sec. 45. Gibbs Statistics.498 Sec. 46. Thermodynamic Quantities.612 Sec. 47. The Thermodynamic Properties of Ideal Gases in Boltzmann Statistics.535 Sec. 48. Fluctuations.646 Sec. 49. Phase Equilibrium.557 Sec. 60. Weak Solutions.568 Sec. 61. Chemical Equilibria.576 Sec. 62. Surface Phenomena.582 Appendix.' . . 586 Bibliography.588 Subject Index.689 FROM THE PREFACE TO THE FIRST EDITION This book is intended for readers who are acquainted with the course of general physics and analysis of nonspecializing institutions of higher education. It is meant chiefly for engineer-physicists, though it may also be useful to specialists working in fields associated with physics—chemists, physical chemists, biophysicists, geophysicists, and astronomers. Like the natural sciences in general, physics is based primarily on experiment, and, what is more, on quantitative experiment. However, no series of experiments can constitute a theory until a rigorous logical relationship is established between them. Theory not only allows us to systematize the available experimental material, but also makes it possible to predict new facts which can be experimentally verified. All physical laws are expressed in the form of quantitative relation¬ ships. In order to interrelate quantitative laws, theoretical physics appeals to mathematics. The methods of theoretical physics, which are based on mathematics, can be fully mastered only by those who have acquired a very considerable volume of mathematical knowledge. Nevertheless, the basic ideas and results of theoretical physics are readily comprehensible to any reader who has an understanding of differential and integral calculus, and is acquainted with vector algebra. This is the minimum of mathematical knowledge required for an understanding of the text that follows. At the same time, the aim of this book is not only to give the reader an idea about what theoretical physics is, but also to furnish him with a working knowledge of the basic methods of theoretical physics. For this reason it has been necessary to adhere, as far as possible, to a rigorous exposition. The reader will more readily agree with the conclusions reached if their inevitability has been made obvious to him. In order to activize the work of the student, some of the applications of the theory have been shifted into the exercises, in which the line of reasoning is not so detailed as in the basic text. In compiling such a relatively small book as this one it has been necessary to cut down on the space devoted to certain important 8 FROM THE PREFACE TO THE FIRST EDITION sections of theoretical physics, and omit other branches entirely. For instance, the mechanics of solid media is not included at all since to set out this branch, even in the same detail as the rest of the text, would mean doubling the size of the book, A few results from the mechanics of continuous media are included in the exercises as illustrations in thermodynamics. At the same time, the mechanics and electrodynamics of solid media are less related to the fundamental, gnosiological problems of physics than microscopic electrodynamics, quantum theory, and statistical physics. For this reason, very little space is devoted to macroscopic electrodynamics: the material has been selected in such a way as to show the reader how the transition is made from microscopic electrod3mamics to the theory of quasi¬ stationary fields and the laws of the propagation of light in media. It is assumed that the reader is familiar with these problems from courses of physics and electricity. On the whole, the book is mainly intended for the reader who is interested in the physics of elementary processes. These considerations have also dictated the choice of material; as in all nonencyclopaedic manuals, this choice is inevitably somewhat subjective. In compiling this book, I have made considerable use of the excellent course of theoretical physics of L. D. Landau and E. M. Lifshits. This comprehensive course can be recommended to all those who wish to obtain a profound understanding of theoretical physics. I should like to express my deep gratitude to my friends who have made important observations: Ya. B. Zeldovich, V. G. Levich, E. L. Feinberg, V. I. Kogan and V. I. Goldansky. A. Kompaneyets PREFACE TO THE SECOND EDITION In this second edition I have attempted to make the presentation more systematic and rigorous without adding any difficulties. In order to do this it has been especially necessary to revise Part III, to which I have added a special section (Sec. 30) setting out the general principles of quantum mechanics; radiation is now considered only with the aid of the quantum theory of the electromagnetic field, since the results obtained from the correspondence principle do not appear sufficiently justified. Gibbs’ statistics are included in this edition, which has made it necessary to divide Part IV into something in the nature of two cycles: Sec. 39-44, where only the results of combmatorial analysis are set out, and Sec. 45-52, an introduction to the Gibbs’ method, which is used as background material for a discussion of thermo¬ dynamics. A phenomenological approach to thermodynamics would nowadays appear an anachronism in a course of theoretical physics. In order not to increase the size of the book overmuch, it has been necessary to omit the theory of beta decay, the variational properties of eigenvalues, and certain other problems included in the first edition. I am greatly indebted to A. F. Nikiforov and V. B. Uvarov for pointing out several inaccuracies in the first edition of the book. A. Kompaneyds PART I MECHANICS Soc. 1. Generalized Coordinates Frames ol reference. In order to describe the motion of a mechanical system, it is necessary to specify its position in space as a function of time. Obviously, it is only meaningful to speak of the relative position of any point. For instance, the position of a flying aircraft is given relative to some coordinate system fixed with respect to the earth; the motion of a charged particle in an accelerator is given relative to the accelerator, etc. The system, relative to which the motion is described, is called a frame of reference. Specification of time. As will be shown later (Sec. 20), specification of time in the general case is also coimected with defining the frame of reference in which it is given. The intuitive conception of a uni¬ versal, unique time, to which we are accustomed in everyday life, is, to a certain extent, an approximation that is only true when the relative speeds of all material particles are small in comparison with the velocity of light. The mechanics of such slow movements is termed Newtonian, since Isaac Newton was the first to formulate its laws. Newton’s laws permit a determination of the position of a mechanical system at an arbitrary instant of time, if the positions and velocities of all points of the system are known at some initial instant, and also if the forces acting in the system are known. Degrees of freedom of a mechanical system. The number of inde-i pendent parameters defining the position of a mechanical system in space is termed the number of its degrees of freedom. The position of a particle in space relative to other bodies is defined with the aid of three independent parameters, for example, its Cartesian coordinates. The position of a system consisting of N particles is determined, in general, by ZN independent parameters. However, if the distribution of points is fixed in any way, then the number of degrees of freedom may be less than 3iV. For example. 12 MECHANICS [Part I if two points are constrained by some form of rigid nondeformable coupling, then, upon the six Cartesian coordinates of these points, yi> * 2 > ^ 2 > imposed the condition (x^ — xi)’‘-h {y^ — + (za — 2i)® = , (1.1) where is the given distance between the points. It follows that the Cartesian coordinates are no longer independent parameters: a relationship exists between them. Only five of the six values Xi, ..., Za are now independent. In other words, a system of two particles, separated by a fixed distance, has five degrees of freedom. If we consider three particles which are rigidly fixed in a triangle, then the coordinates of the third particle must satisfy the two equations; + (2/3 + (Z3 - = Rl, , (1.2) (Xi — XzY + {y^ — y^Y + {z^ — z^)^^R\^. (1.3) Thus, the nine coordinates of the vertices of the rigid triangle are defined by the three equations (1.1), (1.2) and (1.3), and hence only six of the nine quantities are independent. The triangle has six degrees of freedom. The position of a rigid body in space is defined by three points which do not lie on the same straight line. These three points, as we have just seen, have six degrees of freedom. It follows that any rigid body has six degrees of freedom. It should be noted that only such motions of the rigid body are considered as, for example, the rotation of a top, where no noticeable deformation occurs that can affect its motion. Generalized coordinates. It is not always convenient to describe the position of a system in Cartesian coordmates. As we have already seen, when rigid constraints exist, Cartesian coordinates must satisfy supplementary equations. In addition, the choice of coordinate system is arbitrary and should be determined primarily on the basis of expediency. For instance, if the forces depend only on the distances between particles, it is reasonable to introduce these distances into dynamical equations explicitly and not by means of Cartesian coordinates. In other words, a mechanical system can be described by coordinates whose number is equal to the number of degrees of freedom of the system. These coordinates may sometimes coincide with the Cartesian coordinates of some of the particles. For example, in a system of two rigidly connected points, these coordinates can be chosen in the following way; the position of one of the points is given in Cartesian coordinates, after which the other point will always be situated on a sphere whose centre is the first point. The position of the second point on the sphere may be given by its longitude and latitude. Sec. 2] liAGBANGE’S EQUATION 13 Together with the three Cartesian coordinates of the first point, the latitude and longitude of the second point completely define the position of such a system in space. For three rigidly bound points, it is necessary, in accordance with the method just described, to specify the position of one side of the triangle and the angle of rotation of the third vertex about that side. The independent parameters which define the position of a mechanical system in space are called its generalized coordinates. We will represent them by the symbols qa., where the subscript a signifies the number of the degree of freedom. As in the case of Cartesian coordinates, the choice of generalized coordinates is to a considerable extent arbitrary. It must be chosen so that the dynamical laws of motion of the system can be formulated as conveniently as possible. Sec. 2. Lagrange’s Equation In this section, equations of motion will be obtained in terms of arbitrary generalized coordinates. In such form they are especially convenient in theoretical physics. Newton’s Second Law. Motion in mechanics consists in changes in the mutual configuration of bodies in time. In other words, it is described in terms of the mutual distances, or lengths, and intervals of time. As was shown in the preceding section, all motion is relative; it can be specified only in relation to some definite frame of refer¬ ence. In accordance with the level of knowledge of his time, Newton regarded the concepts of length and time interval as absolute, which is to say that these quantities are the same in all frames of reference. As will be shown later, Newton’s assumption was an approximation (see Sec. 20). It holds when the relative speeds of aU the particles are small compared with the velocity of light; here Newtonian mechanics is based on a vast quantity of experimental facts. In formulating the laws of motion a very convenient concept is the material particle, that is, a body whose position is completely defined by three Cartesian coordinates. Strictly speaking, this idealization is not applicable to any body. Nevertheless, it is in every way reasonable when the motion of a body is sufficiently well defined by the displacement in space of any of its particles (for example, the centre of gravity of the body) and is independent of rotations or deformations of the body. If we start with the concept of a particle as the fundamental entity of mechanics, then the law of motion (Newton’s Second Law) is formulated thus: 14 MECHANICS [Part I m d*r d«» F. ( 2 . 1 ) Here, F is the resultant of all the forces applied to the particle fl% p (the vector sum of the forces) is the vector acceleration, the Cartesian components of which are d}x d^y d^z Tt^’ TF’ ~d^' The quantity m involved in equation (2.1) characterizes the particle and is called its mass. Force and mass. Equality (2.1) is the definition of force. However, it should not be regarded as a simple identity or designation, be¬ cause (2.1) establishes the form of the mteraction between bodies in mechanics and thereby actually describes a certain law of nature. The interaction is expressed in the form of a differential equation that includes only the second derivatives of the coordinates with respect to time (and not derivatives, say, of the fourth order). In addition, certain limiting assumptions are usually made in relation to the force. In Newtonian mechanics it is assumed that forces depend only on the mutual arrangement of the bodies at the instant to which the equality refers and do not depend on the con¬ figuration of the bodies at previous times. As we shall see later (see Part II), this supposition about the character of interaction forces is valid only when the speeds of, the bodies are small compared with the velocity of light. The quantity m in equality (2.1) is a characteristic of the body, its mass. Mass may be determuied by comparing the accelerations which the same force imparts to different bodies; the greater the acceleration, the less the mass. In order to measure mass, some body must be regarded as a standard. The choice of a standard body is completely independent of the choice of standards of length and time. This is what makes the dimension (or unit of measurement) of mass a special dimension, not related to the dimensions of length and time. The properties of mass arc established experimentally. Firstly, it can be shown that the mass of two equal quantities of the same substance is equal to twice the mass of each quantity. For example, one can take two identical scale weights and note that a stretched spring gives them equal accelerations. If we join two such weights and subject them to the action of the same spring, which has been stretched by the same amount as for each weight separately, the acceleration will be found to be one half what it was. It follows that the overall mass of the weights is twice as great, since the force depends only on the tension of the spring and could not have changed. Sec. 2] Lagrange’s equation 15 Thus, mass is an additive quantity, that is, one in which the whole is equal to the sum of the quantities of each part taken separately. Experiment shows that the principle of additivity of mass also applies to bodies consisting of different substances. In addition, in Newtonian mechanics, the mass of a body is a constant quantity which does not change with motion. It must not be forgotten that the additivity and constancy of masses are properties that follow only from experimental facts which relate to very specific forms of motion. For example, a very important law, that of the conservation of mass in chemical transformations involving rearrangement of the molecules and atoms of a body, was established by M. V. Lomonosov experimentally. Like all laws deduced from experiment, the principle of additivity of mass has a definite degree of precision. For such strong interactions as take place in the atomic nucleus, the breakdown of the additivity of mass is apparent (for more detail see Sec. 21). We may note that if instead of subjecting a body to the force of a stretched spring it were subjected to the action of gravity, then the acceleration of a body of double mass would be equal to the acceleration of each body separately. From this we conclude that the force of gravity is itself proportional to the mass of a body. Hence, in a vacuum, in the absence of air resistance, all bodies fall with the same acceleration. Inertial frames of reference. In equation (2.1) we have to do with the acceleration of a particle. There is no sense in talking about acceleration without stating to which frame of reference it is referred. For this reason there arises a difficulty in stating the cause of the acceleration. This cause may be either interaction between bodies or it may be due to some distinctive properties of the reference frame itself. For example, the jolt which a passenger experiences when a carriage suddenly stops is evidence that the carri^e is in nonuniform motion relative to the earth. Let us consider a set of bodies not affected by any other bodies, that is, one that is sufficiently far away from them. We can suppose that a frame of reference exists such that all accelerations of the set of bodies considered arise only as a result of the interaction between the bodies. This can be verified if the forces satisfy Newton’s Third Law, i.e., if they are equal and opposite in sign for any pair of particles (it is assumed that the forces occur instantaneously, and this is true only when the speeds of the particles are small compared with the speed of transmission of the interaction). A frame of reference for which the acceleration of a certain set of particles depends only on the interaction between these particles is called an inertial frame (or inertial coordinate system). A free particle, not subject to the action of any other body, moves, relative to such a reference frame, uniformly in a straight line or, in everyday 16 MECHANICS [Part I language, by its own momentum. If in a given frame of reference Newton’s Third Law is not satisfied we can conclude that this is not an inertial system. Thus, a stone thrown directly downwards from a tall tower is deflected towards the east from the direction of the force of gravity. This direction can be independently established with the aid of a suspended weight. It follows that the stone has a component of acceleration which is not caused by the force of the earth’s attraction. From this we conclude that the frame of reference fixed in the earth is noninertial. The noninertiality is, in this case, due to the diurnal rotation of the earth. On the lorccs ol friction. In everyday life we constantly observe the action of forces that arise from direct contacts between bodies. The sliding and rolling of rigid bodies give rise to forces of friction. The action of these forces causes a transition of the macroscopic motion of the body as a whole into the microscopic motion of the constituent atoms and molecules. This is perceived as the generation of heat. Actually, when a body slides an extraordinarily complex ])rocess of interaction occurs between the atoms in the surface layer. A description of this interaction in the simple terms of frictional forces is a very convenient idealization for the mechanics of macro- sco])io motion, but, naturally, does not give us a full picture of the ])roce8s. The concept of frictional force arises as a result of a certain averaging of all the elementary interactions which occur between bodies in contact. In this part, which is concerned only with elementary law's, we shall not consider averaged hiteractions where motion is transferred to the internal, microscopic, degrees of freedom of atoms and molecules. Here, wo will study only those interactions which can be completely expressed with the aid of elementary laws of mechanics and which do not require an appeal to any statistical concepts connected with internal, thermal, motion. Ideal rigid constraints. Bodies in contact also give rise to forces of interaction which can bo reduced to the kinematic properties of rigid constraints. If rigid constraints act in a system they force the particles to move on definite surfaces. Thus, in Sec. 1 we con¬ sidered the motion of a single particle on a sphere, at the centre of which was another particle. This kind of mteraction between particles does not cause a transition of the motion to the internal, microscopic, degrees of freedom of bodies. In other words, motion which is limited by rigid constraints is completely described by its own macroscopic generalized co¬ ordinates 5 *. If the limitations imposed by the constraints distort the motion, they thereby cause accelerations (cimvilinear motion is alw'ays accelerated motion since velocity is a vector quantity). This ac- Sec. 21 Lagrange’s equation 17 celeration can be formally attributed to forces which are called reaction forces of rigid constraints. Reaction forces change only the direction of velocity of a particle but not its magnitude. If they were to alter the magnitude of the velocity, this would produce a change also in the kinetic energy of the particle. According to the law of conservation of energy, heat would then be generated. But this was excluded from consideration from the very start. To summarize, the reaction forces of ideally rigid constraints do not change the kinetic energy of a system. In other words, they do not perform any work on it, shice work performed on a system is equivalent to changing its kinetic energy (if heat is not gener¬ ated). In order that a force should not perform work, it must be jierpen- dicular to the displacement. For this reason the reaction forces of constraints are perpendicular to the direction of particle velocity at each given instant of time. However, in problems of mechanics, the reaction forces are not initially given, as are the functions of particle ijosition. They are determined by integrating equations (2.1), with account taken of coiLstraint conditions. Therefore, it is best to formulate the equations of mechanics so as to exclude constraint reactions entirely. It turns out that if we go over to generalized coord mates, the number of which is equal to the number of degrees of freedom of the system, then the constraint reactions disappear from the equations. In this section wo shall make such a transition and will obtain the equations of mechanics in terms of the generalized coordinates of the system. The transformation from rectangular to generalized coordinates. We take a system with a total of 3iV=w Cartesian coordinates of which V are independent. We will always denote Cartesian coordinates by the same letter Xi, understandmg by this symbol all the co¬ ordinates x,y,z; this means that i varies from 1 to 3A, that is, from 1 to n. The generalized coordinates we denote by q^, (lXi . + dxi (1(1 Oi The total derivative in the first term is written as usual; d I dxi \ ()^ X, dt \8q • • •> q[«> • ■ ^v)' 4) The velocities are substituted in the expression for kinetic energy T= l-2Jmixf, i = l which is now a function of g* and q^. It is essential that in generalized coordinates, T is a function both of and q^. 5) The partial derivatives and are found. * ' ^ oqa oqa. 6) Lagrange’s equations (2.21) are formed according to the number of degrees of freedom. In the next section we will consider some examples in forming Lagrange’s equations. Exercises 1) Write down Lagrange’s equation, where the Lagrangian function has the form: L = —Vi — g* + gg • 2) A point moves in a vortical plane along a given curve in a gravitational field. The equation of the curve in parametric form is x — x (a), z=z (a). Write down Lagrange’s equations. v The velocities are . dx , X = —f— a — X a, da . dz , z — a ^ z a . da The Lagrangian has the form: L = Lagrange’s equation is ^ (x'2 + 2 ' 2 ) gi —'rngz (a) . m [{x'^ + z'*) s] — jn a* {x'x" + z'z") + mgz' = 0 . 24 MECHANICS [Part I Sec. 3. Examples ol Lagrange’s Equations Central forces. Central forces is the name given to those whose directions are along the lines joining the particles and which depend only on the distances between tlicm. Corresponding to such forces, there is always a potential energy, U, dependent on these distances. As an example, wo consider the motion of a particle relative to a fixed centre and attracting it according to Newton’s law. We shall show how to find the potential energy in this case by proceeding from the expression for a gravitational force. Gravitational force is known to be inversely proportional to the square of the distance between the jiarticles and is directed along the line joining them; ( 3 . 1 ) Here a is the factor of proportionality which we will not define more precisely at this point, r is the distance between the particles, and ^ is a unit vector. The minus sign signifies that the particles attract each other, so that the force is in the opposite direction to the radius vector r. According to (3.1), the attractive-force component along x is equal to ( 3 . 2 ) since x is a component of r. But r='Vx'^+y'^-\-z^ » so that and similarly for the tivo other component forces. Comparing (3.3) and (2.7), we see that in the given case = ( 3 - 4 ) We note that the potential energy U is chosen here in such a way that U ( oo) = 0 when the particles are separated by an infinite distance. The choice of the arbitrary constant in the potential energy is called its gauge. In this case it is convenient to choose this constant so that the potential energy tends to zero at infinity. It is obvious that an expression similar to (3.4) is obtained for two electrically charged particles interacting in accordance with Coulomb’s law. Spherical coordinates. Formula (3.4) suggests that in this instance it is best to choose precisely r as a generalized coordinate. In other words, we must transform from Cartesian to spherical coordinates. The relationship between Cartesian and spherical coordinates is Soc. 3] EXAMPLES OF LAGRAXGE’S EQUATIONS shown in Fig. 1. The z-axis is called the polar axis of the spherical coordinate system. The angle S- between the radius vector and the polar axis is called the polar angle; it is complementary (to 90°) to the “latitude.” Finally, the angle 9 is analogous to the “longitude” and is called the azimuth. It measures the diliedral angle between the plane zOx and the plane passing through the polar axis and the given point. Let us find the formulae for the transformation from Cartesian to spherical coordinates. From Fig. 1 it is clear that z^rcosO-. (3.5) The projection p of the radius vector onto the plane xOy is p = rsin.9'. (3.6) Whence, 2 :=p C 0 S 9 = r sin 3- cos 9 , (3.7) 7 / = p sin 9 =r sin 3- sin 9 . (3.8) We wiU now find an expression for the kinetic energy hi spherical coordinates. This can be done either by a simple geometrical con¬ struction or by calculation according to the method of Sec. 2 . Fig. 2 Although the construction is simpler, let us first follow the comjni- tation procedure in order to illustrate the general method. We have: z =f cos 3- —'r sin 3- 3, x—r sin 3 cos 9 -fr cos 3 cos 93 — r sin 3 sin 99 , y—ir sin 3 sin 9 -fr cos 3 sin 93 -fr sin 3 cos 99 . Squaring these equations and adding, we obtain, after very simple manipulations, the following: 26 MECHANICS [Part I T= - -m(x^ + y* + 2^) = + r^^^ + r^sin^&ip*). (3.9) The same is clear from the construction shown in Fig. 2. An arbitrary displacement of the point can be resolved into three mutually per¬ pendicular displacements: dr, rd^ and pd^=r sin 9d2 U(r). (3.11) Now in order to write down Lagrange’s equations it is sufficient to calcidate the partial derivatives. We have: dL ■ 8Ij « a 8L 2 • 2 a • —- = mr , —7- = mr* 0, = mr* sin* 0 6; '
*+-mr0*- dr dr 4^ = wr*sinOcos9' ^ c ?9 These derivatives must be substituted into (2.21), which, however, we will not now do since the motion we are considering actually reduces to the jilane case (see beginning of Sec. 5). Two-particle system. So far we have considered the centre of at¬ traction as stationary, which corresponds to the assumption of an infinitely large mass. In the motion of the earth around the sun, or of an electron in a nuclear field, the mass of the centre of attraction is indeed large compared with the mass of the attracted particle. But it may happen that both masses are similar or equal to each other (a binary star, a neutron-proton system, and the like). We shall show that the problem of the motion of two masses interacting only with one another can always be easily reduced to a problem of the motion of a single mass. Let the mass of the first particle be and of the second mg. We call the radius vectors of these particles, drawn from an arbitrary origin, Tj and Tg, respectively. The components of are x^, y-^, Zy, the components of Tg are Xg, j/g, Zg. We now define the radius vector of the centre of mass of these particles R by the following formula: R = Wi r, -f wtg Tg TOj -f (3.12) Synonymous terms for the “centre of mass” are the “centre of gravity” and the “centre of inertia.” Sec. 3] EXAMPLES OF LAGBANGE’S EQUATIONS 27 In addition, let us introduce the radius vector of the relative position of the particles r=rj,—rg (3.13) Let us now express the kinetic energy in terms of R and r. Prom (3.12) and (3.13) we have r - R 1 ^ mi+rrii ’ (3.14) r3 = R - (3.15) The kinetic energy is equal to y ^ r* 4- F!2. 2 2 2' (3.16) Differentiating (3.14) and (3.15) with respect to time and substi¬ tuting in (3.16), we obtain, after a simple rearrangement. mi 2 (mi -f m^) (3.17) Tf we introduce the Cartesian components of the vectors R (X, Y, Z) and r (x, y, z), then we obtain an expression for the kinetic energy in terms of Cartesian components of velocity. Since no external forces act on the particles, the potential energy can be a function only of their relative positions: U =U {x, y, z). Thus, the Lagrangian is L ^ ( 1 . + 72 + 22 ) + {i- + y^ + z^) - U {x,y,z) . I'ransition to the centrc-ol-inass system. Let us write down La¬ grange’s equations for the coordinates of the centre of mass. We have <7 A = (mi-f mj) F, = + ex ’ BY ’ BL BZ 0 . Hence, in accordance with (2.21) X=F=Z = 0. These equations can be easily integrated: Z=Xot-t-Zo, F=Fot+Fo, Z=Z^+Z^, (3.18) where the letters with the index 0 signify the corresponding values at the initial time. 28 MECHANICS [Part I Combining the coordinate equations into one vector equation, we obtain It ~ Rq • Thus, the centre of mass moves uniformly in a straight line quite independently of the relative motion of the particles. Reduced mass. If wo now write down Lagrange’s equations for relative motion in accordance wdth (2.21) the coordinates of the centre of mass do not appear. It follows that the relative motion occurs as if it were in accordance with the Lagrangian Z/rcl ■ irii 2 (mi + m 2 ) U(r) (3.19) (where = + z*), formed in exactly the same way as the Lagrangian for a single mass m equal to m — - mjmj mi -f- m2 (3.20) This mass is called the redticed mass. The motion of the centre of mass does not affect the relative motion of the masses. In particular, we can consider, simply, th.at the centre of mass lies at the coordinate origin R = 0. In the case of central forces (for example, Newtonian forces of attrac¬ tion) acting between the particles, the potential energy is simply equal to U {r) [this is taken into consideration in (3.10)], where r —Vx^-[-y^-{-z^. Then, if we describe the relative motion in spherical coordinates, the equations of motion will have the same form as for a single particle moving relative to a fixed centre of attraction. The centre of mass can now be considered as fixed, assuming R = 0. From this, in accordance with (3.14) and (3.15), wo obtain the distance of both masses from the centre of mass: wi'o r m, r ^ . y ■ • Y" _ * ^ Wij H-W2 ^ ^ We see that if one mass is much smaller than the other, m., m-i, then Ti i.e., the centre of mass is close to the larger mass. This is the case for a sun-planet system. At the same time the reduced mass can also be written thus: ni = 1+^ mi (3.21) From here it can be seen that it is close to the smaller mass. That is why the motion of the earth around the sun can be approximately described as if the sun were stationary and the earth revolved about it with its own value of mass, independent of the mass of the sun. Sec. 3J EXAMPLES OP laoe.ange’s equations 29 Simple and compound pendulums. In concluding this section we shall derive the Lagrangian for simple and compound pendulums. The simple plane pendulum is a mass suspended on a flat hinge at a certain point of a weightless rod of length 1. The hinge restricts the swing of the pendulum. Let us assume that swinging occurs in the plane of the paper (Fig. 3). It is clear that such a pend¬ ulum has only one degree of freedom. We can take the angle of deflection of the pendulum from the ver¬ tical 9 as a generalized coordinate. Obviously the veloc¬ ity of particle m is equal to so that the kinetic energy is The potential energy is determined by the height of the mass above the mean position z-~l (1 — cos SO that these values are 7 v, 7i. • ,7v. > 7i) 72> •>7v; Qly 72- • • •> 7v), ; 7i. 72> • > > 7i> 72> • • • > 7v). ; 7i’ 72> • , 7v; 7i> 72. • • •. 7v)- (4.2) (4.3) Sec. 4] CONSERVATION LAWS 31 From the equations (4.3) we see that in any mechanical system described by 2v second-order equations there must be 2 v functions of generalized coordinates, velocities, and time, which remain constant in the motion. These functions are called integrals of motion. It is the main aim of mechanics to determine the integrals of motion. If the form of the function (4.3) is known for a given mechanical .system, then its numerical value can be determined from the initial conditions, that is, according to tlie given values of generalized coordinates and velocities at the initial instant. In the preceding section we obtained the so-called centre-of-mass integrals and Rq (3.18). Naturally, Lagi’ange’s equations cannot be integrated in general form for an arbitrary mechanical system. Therefore the problem of determining the integrals of motion is usually very complicated. But there are certain important integrals of motion which are given directly by the form of the Lagrangian. We shall consider these integrals in the present section. Energy. The quantity is called the total energy of a system. Let us calculate its total deriv¬ ative with respect to time. We have _.. ±L_,- d _ 8L .. _ dJ^ dt dqa. dt 8 qa 8qa 8 ij'a 8t The last three terms on the right-hand side are the derivatives of the Lagrangian L, which, in the general form, depend on q, q and t. In determining S’ and its derivative we have made use of the sum¬ mation convention. The quantity ^ in Lagrange’s equations can be replaced by . The result is, therefore, ^ 8 i di dt (4.5) Consequently, if the Lagrangian does not depend explicitly on time ( B L \ = 01, the energy is an integral of motion. Let us find the con¬ ditions for which time does not appear explicitly in the Lagrangian. If the formulae expressing the generalized coordinates q in terms of Cartesian coordinates x do not contain time explicitly (which corre- a is summed from 1 to v (see See. 2). 32 MECHANICS [Part i spends to constant, time-independent constraints) then the transfor¬ mation from X to q cannot introduce time into the Lagrangian. 3 L Besides, in order that = 0, the external forces must also be independent of time. When these two conditions—constant constraints and constant external forces^—are fulfilled, the energy is an integral of motion. To take a particular case, when no external forces act on the system its energy is conserved. Such a .system is called closed. When frictional forces act inside a closed system, the energy of macroscopic motion is transformed into the energy of molecular microscopic motion. The total energy is conserved in this case, too, tliough the Lagrangian, which involves only the generalized coordi¬ nates of macroscopic motion of the system, no longer gives a complete description of the motion of the system. The mechanical energy of only macroscopic motion, determined by means of such a Lagrangian, is not an integral. We will not consider such a system in this section. Let us now consider the case when our definition of energy (4.4) coincides with another definition, S=T+U. Let the kinetic energy be a homogeneous quadratic function of generalized velocities, as expressed in equation (2.13). For this it is necessary that the con¬ straints should not involve time explicitly, otherwise equation (2.6) would have the form . dxi . dxi 'W ’ where the partial derivative of the function (2.2) with respect to time is taken for all constants q^. But then terms containing q^ in the first degree would appear in the expression for T. Since we assume that the potential energy U does not depend on velocity [see (2.18) and (2.19)], then dL ^ 8T dqa ~ dqa ’ and the energy is ^ = L. (4.6) But according to Euler’s theorem, the sum of partial derivatives of a homogeneous quadratic function, multiplied by the corresponding variables, is equal to twice the value of the function (this can easily be verified from the function of two variables ax^ + 2 bxy+cy^). Thus, ^=:2T-~T+U=T+U, (4.7) that is, the total energy is equal to the sum of the potential energy and the kinetic energy, in agreement with the elementary definition. We note that the definition (4.4) is more useful and general also in the case when the Lagrangian is not represented as the difference Sec. 4] CONSERVATION LAWS 33 L—T — U. Thus, in electrodynamics (Sec. 16) L contains a linear term in velocity. For the energy integi'al to exist, only one condition Q Ij is necessary and sufficient: — 0 (if, of course, there are no friction¬ al forces). The application of the energy integral to systems with one degree of freedom. The energy integral allows us, straightway, to reduce problems of the motion of systems with one degree of freedom to those of quadrature. Thus, in the pendulum problem considered in the previous section we can, with the aid of (4.7), uTite down the energy integral directly: 92=^1 r3> pfc - i = ri—rfe, Then the derivatives ... will be expressed in terms of the variables p^, ..., p), _ i, ... as follows: N-1 k-1 8U_ 8t, 8t, Fi —Fg... ■’ 8r7 1 in (4.23) we obtain N N II k^l N-1 1 N-1 N-1 ? ki 1 _ fc=i fe-i 8U 9 ?S.'-I In this expression, only the relative coordinates p^, remain. We shall now show that, for a closed system, the right-hand side is identically equal to zero. The potential energy is a scalar function of coordinates. Hence, it can depend only on the scalar arguments pfe^i pi^. (p/ pt). totally irrespective of whether the initial expression was a function only of the absolute values | r; — r*, |, or whether it also involved scalar products of the form (rj— Tk, ii —Fn). An essential point is that the system is closed (in accordance with the definition, see page 32), and the forces in it are completely defined by the relative positions of the points and by nothing else. Therefore, the potential energy depends only on the quantities r,-—r*, and only in scalar combinations (r,-— tk, r/ — r„) (in particular, the subscripts i, 1; k, n can even be the same; then the scalar product becomes the square of the distance between the particles i, k). To summarize, the potential energy U depends on the following arguments: u=u [p”, p“,..., pL ..., pN-i; (pi P2).(pfcp/)]. 40 MKC'HANIt'S [Part I In order to save sjiace we will, in future, perform the operation for two vectors, though this 0 ])eration can be directly generalized to any arbitrary number. We obtain 'oU c)pi " dU ~ ^(Pi) ^pi + eu ^(Pl P2) f'(Pi P 2 ) c>Pj dV _ dU 1 eu ^ (Pi P 2 ) «’P2 “ e(pi) ^Pi ^(pi P 2 ) 8p„ The partial derivatives of the scalar quantities p*, (pj pg) witli respect to the vector arguments are in the giveji case easily (waluated. Tims. Spi f!iPjP=l cipi - P 2 - hlach of these equations is a shortened form of three equations referring to the components (the components of p, are y),-, 0): Hence, ;,5-(PiP 2)= (?l?2 + -')l>l2 + ^lQ=?2; (Pi P 2 ) “ ’^ 2 ' (P 1 P 2 ) — ^2 • < Pi ^{PiPi) ’ '-'Pi Pi S(PiPa) (4.25) Substituting (4.25) into (4.24) for the case of the two variables, we obtain = 1 , is i= —(r‘‘+rV)-(^(»-). (5.3) where m is the reduced mass. Angular momentum as a generalized momentum. We shall now show 0 Ij that = Af is nothing other than , i.e., the component of angular momentum along the polar axis is a generalized momentum, provided the angle of rotation 9 around that axis is a generalized coordinate. Indeed, in accordance with (5.2), the angular momentum M is M = Mz=x'py — ypx=r'mr cos 9 (f sm 9 +r cos 99 )— —mr sin 9 (fcos 9 —r sin 99 )=jKr* 9 (cos^ 94 -sin^ 9 )=mr 2 ^ On the other hand, differentiating L with respect to 9 we see that dL 99 = mr®9 = Mz (5.4) The expression for angular momentum in polar coordinates can also be derived geometrically (Fig. 6 ). In unit time, the radius vector r moves to the position shown in Fig. 6 by the dashed line. Twice the area of the sector OAB, multiplied by the mass m, is by definition equal to the angular momentum [cf. (5.2)]. But, to a first approxi¬ mation the area of the sector is equal to the product of the modulus r and . The height h is proportional to the angle of rotation in unit Sec. 5] MOTION m A CENTRAL FIELD 43 time and to the radius itself so that the area of the sector is 1/2 r* ■) CO tends to zero at infinity faster than, then the integral j F dr is CO r convergent. Then we ean put U (^)=j F dr, or U (oo)—0. In other r words, the potential energy is considered zero at infinity. The elufice of an arbitrary constant in the c.xprcssion for potential energy is called its ijduge. In addition we shall consider that at r —0, U (r) does not tend to infinity more rapidly than , as, for c.xainple, for Newtonian attrac¬ tion U -- -• j | j dr - - - II r Lot us now write (fi.fi) as til. /■“ 2 J\I^ 2mr'^ -U(r). The left-hand side of this equation is essentially positive. For r=oo the last two terms in (.').?) tend to zero. Thus, for the particles to be able to recede from each other an infinite distance, the total energy must be positive when the gauge of the potential energy satisfies f7(oo)=0. (Jiven a dcliiiite form of U, we ean now plot the curve of the function UM(r)^ + U (r) . (.5.8) The index M in U dejiotes that the potential energy includes the IP “centrifugal” eiiergy The derivative of this quantity- with respect to r, taken with the opposite sign, is equal to . If we put il/ = w?r® 9 , the result will be the usual ex])ression for “centrifugal force.” However, henceforth, we will call a mechanical quantity of different origin the “centrifugal force” (see Sec. 8). Let (7<0 and monotonie. Since U (cx)) = 0, we see that U (r) is an increasing function of r. It follows that the force has a negative sign |sinceF=— , i.e., it is an attractive force. Let us assume, in addition, that at infinity \12 I ('■) I Let us summarize the assumptions that we have made concerning Untir): 1 ) f/iU (r) is positively infinite at zero, where the centrifugal term is predominant. 2 ) at infinity, where U (r) predominates, U.u (r) tends to zero from a negative direction. Sec. 5] MOTION IN A CENTRAL FIELD 45 Consequently, the curve Uu^r) has the form shown in Fig. 7, since we must go through a minimum in order to pass from a decrease for small values of r to an increase at large values of r.* In this figure we can also plot the total energy <5". But since the total energy is conserved, the curve of S must have the form of a horizontal straight line lying above or below the abscissa, depending on the sign of 6. For positive values of energy, the line 0, w'hich corresponds to repulsion, the curve Um (r) docs not possess a minimum. Then finite motion is clearly im]iossil)le. Falling towards the centre. For Newtonian attraction, U (0) tends to infinity like —1/r. If we su])pose that U (0) tends to —oo more rapidly than — 1/r^, then the curve UM{r) is negative for all r close to zero. Then, from (5.7), is positive for infinitesimal values of r and tends to infinity when r tends to zero. If f<() initially, then f does not change sign and the particles now begin to move towards collision. In Newtonian attraction this is possible only when the particles are directed towards each other; then “the arm” of the angular momentum is equal to zero and, hence, the angular momentum itself is obviously equal to zero, too, so that Um (r) = U (r). If an initial “arm” exists within the distance of minimum approach, then ♦ rf I {/ {r) |< 2 fnri infinity, then the curve approaches zero on the positive side, and there can bo a further small maximum after the minimum. This form of C/m (r) applies to the atoms of elements with medium and largo atomic weights. MECHANICS 46 [Part 1 the angular momentum M =mvp 0 (p is the “arm”) and the motion can in no way become radial. In the case of Newtonian or Coulomb attraction for a particle with angular momentum not equal to zero, there always exists a distance Tq 'for which v—r becomes greater than S’ — U (ro). This distance A ffh T ^ determines the perihelion for the approaching particles. However, if U (r) tends to infinity more rapidly than — then, as r->0, there will be no 2 >oint at which Um (r) becomes zero. In place of a hyperbolic orbit, as in the case of Newtonian attraction, a siiiral curve leading to one particle falling on the other results. The turns of the spiral diminish, but the speed of rotation increases so that the angular momentum is conserved, as it should be in any central field. But the “centrifugal” reinilsive force turns out to be less than the forces of attraction, and the particles approach each other indef¬ initely. Of course, the result is the same if the energy is negative (for example, part of the energy is transferred to some third particle, which then recedes). In the case of attractive forces increasing more rapidly than 1 /r®, no counteqiart to elli])tical orbits exists. If three bodies in motion are subject to Ncivtonian attraction, two of them may coUide even if, initially, the motion of these particles was not jiurely radial. Indeed, in the case of three bodies, only the total angular momentum is conserved, and this does not exclude the collision of two particles. Ucducing to quadrature. Lot us now find the equation of the tra¬ jectory in general form. To do this we must, in (5.6), change from differentiation with respect to time to differentiation with respect to cp. Using (5.4) we have = (5.9) Se]mrating the variables and passing to 9 in (5.6) gives 1 M dr (5.10) 1 mr^ 1 / ^0 ’ Here, the lower limit of the integral corresponds to 9 = 0 . If we cal¬ culate 9 with respect to the perihelion, then the corresponding value r=rQ can be easily found by noting that the radial component of velocity r changes sign at perihelion (r has a minimum, and so dr = 0 ). From this we find the equation for the particle distance at perihelion: Sec. 5] MOTION IN A CENTBAL FIELD 47 Kepler’s problem. Thus, the problem of motion in a central field is reduced to quadrature. The fact that the integral sometimes cannot be solved in terms of elementary functions is no longer so essential. Indeed, the solution of the problem in terms of definite integrals contains all the initial data explicitly; if these data are knoivn, the integration can be performed in some way or other. But, naturally, if the integral is expressed in the form of a well-known function, the solution can be more easily examined in the general form. In this sense an explicit solution is of particular interest. Such a solution can be found in only a few cases. One of these is the case of a central force diminishing inversely as the square of distance. The forces of Newtonian attraction between poiat masses (or bodies possessing spherical symmetry) are subject to this law. It will be recalled that the laws of motion in this case were found empirically by Kepler before Newton deduced them from the equations of mechanics and the law of gravitation. It was the agreement of Newton’s results mth Kepler’s laws that was the first verification of the truth of Newtonian mechanics. The problem of the motion of a particle in a field of force diminishing inversely with the square of the distance from some fixed point, is called Kepler’s problem. The prob¬ lem of the motion of two bodies with arbitrary masses always reduces to the problem of a single body when passing to a frame of reference fixed in the centre of mass. The expression “Kepler’s problem” can also be applied to Coulomb forces acting between point charges. These can either be forces of attraction or repulsion. In all cases we shall write U = ^, where a < 0 for attractive forces and a>0 for repulsive forces. H we replace — - in (5.10) by a new variable x, the integral in the Kepler problem is reduced to the form dx X -f 2a M m arc cos M ]T~ \ M* -f 2 ^ m M M X — - At the lower limit, the expression inside the arc cos sign is equal to unity [as will be seen from (6.11)], since the lower limit was chosen on the condition that dr — 0. But arc cos 1=0. Rearranging the result of integration and reverting to r, we obtain, after simple manipu¬ lations. am ., 2S' -' + -^1 - COS

0 the eccentricity is greater than unity and the denominator in (6.12) becomes zero for a certain cp = (poo. Thus, the orbit goes to infinity (a hyperbola). The direction of the asymptote is obtained by putting r =oo in (5.12). This requires that cos tpoo = _ ■ The angle between the asymptotes is equal to 2 900, when the particles repel each other, and to 2 (tt— 900), when the particles attract. An example of a trajectory, when the forces are repulsive, is shown in Fig. 8, Sec. 6. Exercise . HT^ Obtain the ocination of the trajectory when U — ^ > 0. See. 6. Collision of Particles The significance of collision problems. In order to determine the forces acting between particles, it is necessary to study the motion of particles caused by these forces. Thus, Newton’s gravitational law was established with the aid of Kepler’s laws. Here, the forces were determined from finite motion. However, infinite motion can also be used if one particle can, in some way, be accelerated to a definite velocity and then made to pass close to another particle. Such a process is termed “collision” of particles. It is not at aU assumed, however, that the particles actually come into contact in the sense of “collision” in everyday life. And neither is it necessary that the incident particle should be artificially accelerated in a machine: it may be obtained in ejection from a radioactive nucleus, or as the result of a nuclear reaction, or it may be a fast particle in cosmic radiation. Two approaches are possible to problems on particle collisions. Firstly, it may be only the velocities of the particles long before the collision (before they begin to interact) that are given, and the problem is to determine only their velocities (magnitude and direction) after they have ceased to interact. In other words, only the result of the collision is obtained without a detailed examination of the process. In this case, some knowledge of the final state must be available (or .Sec. 0] COLIJ.SIOX OF PAKTICLES 49 specified) beforehand: it is not possible to determine, from the initial velocities alone, aU the integrals of motion which characterize the collision, and, hence, it is likewise impossible to predict the final state. With this approach to collision problems, only the momentum and energy integrals are known. However, another aijproach is possible: it is required to precalculate the final state where the precise initial state is given. Let us first consider collisions by the first method. It is clear that if only the initial velocities of the particles are known, the collision is not completely determined: it is not known at what distance the par¬ ticles were when they passed each other. This is why some quantity relating to the final state of the system must be giveir. Usually the problem is stated as follows: the initial velocities of the colliding par¬ ticles and also the direction of velocity of one of them after the colli¬ sion are specified. It is required to determine all the remaining quanti¬ ties after the collision. In such a form the problem is solved uniquely. Six quantities are unkno\vn, namely the six momentum components of both particles after the collision. The conservation laws ])rovide four equalities: conservation of the scalar quantity (energy) and conservation of the three components of the vector quantity (momen¬ tum). Therefore, with six unknowns, it is necessary to specify two quantities which refer to the final state. They are contained in the determination of the unit vector which specifies the direction of the velocity of one of the particles; an arbitrary vector is defined by three quantities, but a unit vector, obviously, only by two. Actually, only the angle of deflection of the particles after the collision need be given, i.e., the angle which the velocity of the particle makes with the initial direction of the incident particle. The orientation (in space) of the plane passing through both velocity vectors is im¬ material. Elastic and inelastic collisions. A collision is termed elastic, if the initial kinetic energy is conserved when the particles separate after the collision at infinity, and inelastic, if, as a result of the collision, the kinetic energy changes at infinity. In nuclear physics, studies are very often made of collisions of a more general character, in which the nature of the colliding particles changes. These collisions are also inelastic. They are called nuclear reactions. The laboratory and centre-o!-mass frames of reference. When colli¬ sions are studied in the laboratory, one of the particles is usually at rest prior to collision. The frame of reference fixed in this particle (and in the laboratory) is termed the laboratory frame. However, it is more convenient to perform calculations in a frame of reference, relative to which the centre of mass of both particles is at rest. In accordance with the law of conservation of the centre of mass (3.18), it will also be at rest in its own frame after the collision. The velocity of the centre of mass, relative to the laboratory frame of reference, is 60 MECHANICS [Part I Y — _ TO 1 + W 2 ( 6 . 1 ) Here Vq is the velocity of the first particle (of mass trij) relative to the second (with mass m 2 ). In so far as the second particle is at rest in the laboratory system, Vq is also the velocity of the first particle relative to this system. The general case of an inelastic collision. The velocity of the first particle relative to the centre of mass is, according to the law of addition of velocities, equal to Vio = Vo in^ y„ __ 4- Wj ’ ( 6 . 2 ) and in the same system, the velocity of the second particle is ^ 20 “■ OTi + «ij ■ (6.3) Thus, mi Vio+mg V 2 o = 0, as it should be in the centre-of-mass system. In accordance with (3.17), the energy in the centre-of-mass system is Vl (6.4) Here, the reduced mass is indicated by a zero subscript, since in nuclear reactions it may change. Let the masses of the particles obtained as a result of the reaction be m 3 and and the energy absorbed or emitted Q (the so-called “heat” of the reaction). If Q is the energy released in radiation, then, strictly speaking, one should take into account the radiated momen¬ tum (see Sec. 13). But it is negligibly small in comparison with the momenta of nuclear particles. Thus, the law of conservation of energy must be written in the follow¬ ing form: movj mv* 2 + V 2 • (6.5) Here, m = is the reduced mass of the particles produced in the nuclear reaction, and v is their relative velocity. In order to specify the collision completely, w'e will consider that the direction of v is known, since the value of v is determined from ( 6 . 6 ). Then the velocity of each particle separately will be ''30 V 40 WaV ma+m* ■ ( 6 . 6 ) They satisfy the requirement Wj V 3 o-f-m 4 V 4 o= 0 , i. e., the law of conservation of momentum in the centre-of-mass system, and give the necessary value for the kinetic energy Sec. 6] COLUSION OP PABTICI.ES 61 ‘^h'^30 , «» 4«40 2 ” 2 Now, it is not difficult to revert to the laboratory frame of reference. The velocities of the particles in this system will be V3=-V3o+V^ OT 4 V ^V40 + V = ~- »h + nif + »n,v„ «ij + Wa ' w,T„ OT 3 + W 4 + nil + »”a (6.7) Equations (6.7) give a complete solution to the problem provided the direction of v is given. Elastic collisions. The computations are simplified if the collision is elastic, for then Q~0. It follows from (6.5) that the relative velocity changes only m direction and not in magnitude. Let us suppose that its angle of deflection x is given. We take the axis Ox along Vq, and let the axis Oy lie in the plane of the vectors v and v^ (Avhich are equal in magnitude in the case of elastic collisions). Then Vx =I'o cos X, Wy=«o sin X • From (6.6), the components of particle velocity in the centre-of-mass system will, after collision, be correspondingly equal to Viox = ^ OTgt>oC08X nil + nil *> 20 * — Wj Vo cos X mi + nil ’ *>20V — Vq sin X mi + Wj ’ miVosinx mi + mi Since the velocity of the centre of mass is in the direction of the axis Ox, we obtain, from (6.7), the equations for the velocities in the laboratory frame of reference: *'i* (mi + miC 08 x) Vq mi + mi Wi (1 — cosx) Vq mi 4- mi ■ Vinv -;■ mjVoSmx Wi + m, *> 2 y — *> 2 oy — — m,VoSinx mj + mj By means of these equations, the deflection angle 0 of the first particle in the case of collision in the laboratory system can be related to the angle x, (i- e., its deflection angle in a centre-of-mass system): ^ '*'iy misinx ® t’lx mi + miooax The “recoil” angle of the second particle 6' is defined as tan0' Viy _ sinx Vix 1 — cos X ( 6 . 8 ) (6.9) 62 MECHANICS [Part I Tlie minus sign in the dciinition of tg 0' is chosen because the signs of v-^y and are opposite. The case of equal masses. Equation (0.8) becomes still simpler if the masses of the collidmg jiarticles are equal. This is approximately true in the case of a collision between a neutron and proton. Then, from ( 6 . 8 ). tan 0 " tan , Q — ~ ^ 6 + 6 ' TT 2 ^ i.e., the particles fly off at right angles and the deflection angle of the neutron in the laboratory system is equal to half the deflection angle in the centre-of-mass system. Since the latter varies from 0 to 180°, 6 cannot exceed 90°. And, in addition, the velocity of the incident particle is plotted as the “resultant” velocity of the diverging particles. The collision of billiard balls resembles the collision considered here of particles of equal mass, provided that the rotation of the balls about their axes is neglected. The energy transferred in an elastic collision. The energy received by the second particle in a collision is CTaWf (l -- -co8x)i>g (TOi + wq)* Its portion, relative to the initial energy of the first particle, is (fj _—oosz) ^0 ^ (mi + Wa)® ( 6 . 10 ) From this we obtain ^ = .sin^ ■— =sin2 6 for particles of equal mass. Accordingly, the jiortion of the energy retained by the fir.st particle is-C-=cos®0. In a “head-on” collision x —180°, 6 = 90°. 00 The first particle comes to rest and the second continues to move forward with the same velocity. This can easily be seen when billiard balls collide. The problem of scalicring. Let us now examine the problem of colli¬ sion in more detail. We shall confine ourselves to the case of elastic collision and perform the calculations in the centre-of-mass system. The transformation to the laboratory system by equations (6.7) is elementary. It is obvious that for a complete solution of the collision problem, one must know the potential energy of interaction between the par¬ ticles U (r) and specify the initial conditions, so that aU the integrals of motion may be determined. The angular-momentum integral is found in the foUowuig way. Fig. 8, which refers to repulsive forces. Sec. 6] COLLISION OF PABTICLES 63 represents the motion of the first particle relative to the second. The path at an infinite distance is linear, because no forces act between the particles at infinity. Since the path is linear at infinity, it possesses asymptotes. The asymptote for the part of the trajectory over which the j)articles are approaching is represented by the straight line AF, and FB is the asymptote for the part whore they recede. The collision parameter. The distance of an asymptote from the straight line OC, drawn through the second point and parallel to the relative velocity of the particles at infinity, is called the collision parmneter (“aiming distance”). It has been denoted ^ by p, since, as can be seen from Fig. 8 , p is also the “arm” of the angular momentum. If there were no interaction between the particles, tiiey would pass each other at a distance p; this is why p is called the collision ])aramoter. lint we know that the angular momentum is very sim])ly expressed in terms of p. In the preceding section it ivas shown that it is equal to mvp. Let us draw the radius vector OA to some very distant jioint A. Then the angular momentum is M = mvr sin a (the angle a is shown on the diagram). But rsina = p, so that M =mDp . (fi ll) Recall that here m is the reduced mass of the particles and v is their relative velocity at infinity. The energy integral is expressed in terms of the velocity at infinity thus: ( 6 . 12 ) since U (oo) — 0. The deflection angle. The deflection angle y is equal to | tt —29 „ |, where 9 ^. is half the angle between the asymptotes. The angle 9 „ corresponds to a rotation of the radius vector from the position OA, where it is infinite, to a position OF, where it is a minimum. Hence, from equation (5.10) the angle 900 is expressed as (6.13) rg is determined from (5.11). In place of M and S' we must substitute into (6.13) the expressions (fi.ll) and (6.12). 54 MKCHANICS [Part I The differential effective scattering cross-section. Let us suppose that the integral (6.13) lias been calculated. Tlien and therefore Xj are known as functions of the collision parameter p. Let this relation¬ ship bo inverted, i.e., p is determined as a function of the deflection angle: p = p(x). (6.14) In collision experiments, the collision parameter is never defined in practice; a parallel beam of scattering particles is directed with identical velocity at some kind of substance, the atoms or nuclei of which are scatterers. The distribution of particles as to deflection angles x (or> more exactly, as to angles 0 in the laboratory system) is observed. Thus a scattering experiment is, as it were, performed very many times one after the other with the widest range of aiming distances. Tjct one particle pass through a square centimetre of surface of the scattering substance. Then, in an amiulus contained between p and p-[-dp, there pass ‘Inp dp particles. We classify the collisions according to the aiming distances, similar to the way that it is done on a shooting target with the aid of a concentric system of rings. If p is known in relation to x, then it may be stated that da=2np dp = = 2Tzp-~dx particles will be deflected at the angle between x and x + dx- Let us suppose that the scattered particles are in some way detected at a large distance from the scattermg medium. Then the whole scatterer can be considered as a point and we can say that after scattering the particles move in straight lines from a common centre. Let us consider those particles which occupy the space between two cones that have the same apex and a common axis ; the half-angle of the imier cone is equal to x> the external cone x+<^X- The space between the two cones is called a solid angle, similar to the way that the plane contained between two straight lines is called a plane angle. The measure of a plane angle is the arc of a circle of unit radius drawn about the vertex of the angle, while the measure of a solid angle is the area of a sphere of unit radius dra^vn about the centre of the cone. An elementary solid angle is shown in Fig. 9 as that part of the surface of a sphere covered by an element of arc dy when it is rotated about the radius OC. Since 00—1, the radius of rotation of the element dx is equal to sin x- Therefore, the surface of the sphere which it covers is equal to 2n sin xdx- Thus, the elemen¬ tary solid angle is Sec. 6] COLiaSIOK OF PARTICLES 65 dOi = 27 t sill y^dx- (6.15) The number of particles scattered in the element of solid angle is, thus, da = p dp do dx sinx (6.16) The quantity da has the dimensions of area. It is the area in which a particle must fall in order to be scattered within the solid-angle element dQ. It is called the effective differential scattering cross-section in the element of solid angle dQ. Experimentally we determine just this value, because it is the angular distribution of the scattered particles that is dealt with [in (6.16) we consider that p is given in relation to /]. If there are n scatterers in unit volume of the scattering substance, then the attenu¬ ation of the primary beam J in passing through unit thickness of the substance, due to scattermg in an elementary solid angle d£i, is dJa — — Jnda= — Jnp particles/cm. If we examine da as a function of y, we find a relationship between the collision parameter and the deflection angle. And this allows us to draw certain conclusions about the nature of the forces acting between the particle and the scattering centre. Rutherlord’s formula. A marvellous example of the determination of forces from the scattering law is given by the classical experiments of Rutherford with alpha particles. As was pointed out in Sec. 3, the Coulomb potential acting on particles decreases with distance according to a y law, in the same way as the Newtonian potential. Consequently, the deflection angle can be calculated from the equations of Sec. 5. Let us first of aU find the angle 0 (the charges on the nucleus and alpha particle are like charges). Hence, COS(poo = 1 2M^ ¥ ’ ma^ tan 9oo = M 1/2^ aim' (6.17) The integrals of motion S and M are determined with the aid of (6.11) and (6.12). We therefore have p =tg

oo. Sec. 7] SMAI.L OSCILLATIONS 57 and if r>rQ, then U (»•)->- 0 . When n = oo the scattering is completely isotropic. If n is large, the angular distribution of the particle is almost isotropic, and only for very small deflection angles has the distribution a sharp maximum. Hence, a scattering law that is almost isotropic indicates a rapid diminution of force with distance. The scattering of neutrons by protons in the centre-of-mass system is isotropic up to energy values greater than 10 Mev (1 Mev equals 1.6 X 10“® erg). An analysis of tlie effective cross-section sliows that nuclear forces are short-range forces; they are very great at close distances and rapidly diminish to zero at distances larger than 2 X 10 - 1 ® cm must be mentioned, however, that a correct investi¬ gation of this case is only possible on the basis of the quantum theory of scattering (Sec. 37). Exercises 1) Find the differential effective scattering cross-section for particles by an impermeable sphere of radius r„. The impermeable splioro can be represented by giving the potential energy in the form U (r) = 0 for r > (outside the sphere) and V (r) -- oo for r < (inside the siihcrc). Then, whatever the kinetic energy of tho particle, penetra¬ tion into the region r 1 with tho initial flight direction of the particle of mass mi- Determine the energy Q which is absorbed or emitted in the collision. Sec. 7. Small Oscillations In applications of mechanics, we very often meet a special form of motion known as small oscillations. We devote a separate section to the theory of small oscillations. 68 MECHANICS [Part I The definition ot small oscillations of a pendulum. In the problem of pendulum oscillations in Sec. 4 it was shown that the equation relating the deflection angle 9 to time led, in the general case, to a nonelementary (elliptical) integral (4.11). A simple graphical in¬ vestigation shows that the function 9 (t) is periodic. Fig. 10 shows the curve U ( 0 . (7.17) Under this condition, U has a minimum at the point gi = 0, g 2 = 0. Let us rewrite the left-hand side of (7.17) in identical form + 2Pi2?i? 2 + P2a?l) = -^(?x + + -^^f^^-g|. This expression remains positive for all values of g^ and g 2 , provided the coefficients of both quadratics in g are greater than zero; Pu>0. (7-18) Pup22-P?2>0. (7.19) In future, we shall consider that the conditions (7.18) and (7.19), together with analogous conditions for a^, and a**, are satisfied. We shall now write down Lagrange’s equations. We have *ii9i+*i 2 ^/a> *ia9 + “aa^a> = Pii?i + Pi*?* > 8^~ Paaffa • Whence aii?i + «i2?2+Pu'i'i + Piag'a = 0. 1 ^,^20) *I2?l+a22?2+Pia3'l+p22 32 = 9. j In order to satisfy these equations, we shall look for a solution in the form gi=^ie*“', g2=J2e‘'“'. (7.21) As in (7.14c), the real part of the solution (7.21) must be taken. Sec. 7] SMALL OSCILLATIONS 63 The equation for frequency. Substituting (7.21) in (7.20), we obtain equations relating and A 2 ' (Pll —«llW^)^I + (Pl 2 —«I 2«^)^2 = 0 , I (P 12 *12 -^I d" (P 22 *22 “‘^ 2 “® • i Transferring terms in .dg to the right-hand side of the equation and dividing one equation by the other, ive eliminate Aj^ and .dg: Pil —«ii _ P 12 —g iao ’* Pl 2 -ai 2 ^22 - «22 (7.23) Reducing (7.23) to a common denominator, we arrive at the bi¬ quadratic equation (*n*22 *12) (Pll*22+ ^22*11 ^a^gpig) + PuP 22 -Pf 2 = 0. (7.24) Substituting here the expressions for m, from (7.15), wo obtain an equation for the frequencies of a double pendulum — (m-f TOj) nijllx (^i-t-^) m^) mJlxg^ — O. If we introduce stiU another contraction in notation (for the given problem) -^ = 7^, = P, the expression for frequencies will be of the following form: = W + P) (1 + X) ± V (1 -f (x)2 (1 -f X)2 — 4X (1+ p) ]. It is easy to see that this expression yields only the real values of the frequencies. However, we shall show this in more general form for equation (7.24). Let us assume that the following function is given: F (t>)^)= (a^^agg afg) w* (Pii*22 d" ^22*11 ^p^ga^g) w®-!- P11P22 which passes through zero for all values of <0 that satisfy equation (7.24) . jP (w®) is positive for w*=0 and for w® = oo, since Pupgg — — P 12 > 0, aiiagg — afg > 0. Let us now substitute into this function the positive number {>>*==-|^. After a simple rearrangement we P 22 obtain *11 ^ l"^) ~ ^12 *22)* ® • Thus, as CO® varies from 0 to oo, (co®) is first positive, then negative, and then again positive. Hence, it changes sign twice, so that equation (7.24) has two positive roots cof, (o“ and, as was asserted, all the values for frequency are real. 64 MECHANICS [Part I The quantity co has four values, both pairs of which are equal in absolute value. If we represent the solution in the form (7.21), it is sufficient to take only positive w. Normal coordinates. Let us put these roots in (7.22). To each of them there will correspond a definite ratio of tlie coefficients For i = l, 2 we have 1 Pll — «u“i Pl 2 “12“," (i=l, 2 ). (7.25) According to (7.23), the same ratio is also obtained from the second equation of (7.22). For example, for the double pendulum '. (7.26) We must, of course, take oidy the real parts of the expressions on the right. We now introduce the following notation: s= = (? 2 . (7.27) According to (7.27), the quantities and satisfy the differential equations 4 + ‘^iei = 0: 4 + wi^)2 = 0. (7.28) Each of these equations can be obtained from the Lagrangian 4 —[wfQ?, (7.29) which describes oscillation with one degree of freedom. Thus, in terms of the variables Qi, the problem of two related oscillations with two degrees of freedom q^, has been reduced to the problem of two independent harmonic oscillations with one degree of freedom and Q^. The coordinates and Q.^ are termed normal. In equations (7.20), we cannot arbitrarily put fg?), (7.31) 1 since L = T — U and S' —T+U. This result is true for small oscil¬ lations with any number of degrees of freedom. We must note that if the normal coordinates are expressed directly by equations (7.30), then the separate energy terms — (Q?co?Qf), will also be multiplied by certain numbers a,-. However, if we replace Qi by Qi V^i, then these numbers are eliminated from the expression for energy, which is then reduced to the form (7.31). An example of this procedure is given in the exercises. Thus, the energy of any system performing small oscillations is reduced to the sum of the energies of separate, independent linear harmonic oscillators. As a result of this, consideration of oscillation problems is greatly simplified since the linear harmonic oscillator is, in many respects, one of the most simple mechanical systems. The reduction to normal coordinates turns out to be a very fruitful method in studies of the oscillations of polyatomic molecules, in the theory of crystals, and in electrodynamics. In addition, normal coordinates are useful in technical applications of oscillation theory. The case ol equal frequencies. If the roots of equation (7.24) coincide, the general solution must not be written in the form (7.26), but somewhat differently, namely, qi=A cos (at -\-B sin r])“ - (r). (8.5) Let us now write down Lagrange’s equations for motion relative to a rotating system, i.e., considering r a generalized coordinate. In order to do this we must calculate the derivatives and-^ ; or or let it be noted that differentiation with respect to a vector denotes a shortened way of writing down the differentiation with respect to all of its three components. The general rules for such differentiations will be given in Sec. 11; here we shall calculate the derivatives for each component separately. Let w be along the direction of the z-axis. Then, in vector com¬ ponents, L will be of the form L=^[(x—yy+(y + toa:)* + z^'\ — U{x,y,z ). (8.6) Whence we obtain -^ = m(x —coy), = m (y-f coa:), -^ = mz\ -^ = m] + 2m [rw] + m [to [rco]] -(8.7) Expanding the double vector product on the right by means of the equation [A [BC]]=B (AC) — 0 (AB), and transforming to com¬ ponents, we can see that (8.7) is equivalent to the preceduig system of three equations. A direct differentiation with respect to the vectors r and f would have led to (8.7), without the expression in terms of components. Inertial forces. The first three terms on the right in (8.7) essentially distinguish the equations of motion, written relative to a rotating coordinate system, from the equations written relative to a non¬ rotating system. The use of a noninertial system is determined by the nature of the problem. For example, if the motion of terrestrial bodies is being studied, it is natural to choose the earth as the coordinate system, and not some other system related to the Galaxy (the aggregate of stars in the Milky Way). If we consider the reaction of a passenger to a train that suddenly stops, we must take the train as frame of reference and not the station platform. When the train is braked sharply, the passenger continues to move forwards “inertially” or, as we have agreed to say, he continues to move uniformly relative to an inertial system attached to the earth. Thus, relative to the carriage, it is the familiar jerk forward. At the same time it is obvious that the noninertial system is the train and not the earth, since no one experiences any jerk on the platform. The additional terms on the right of equation (8.7) have the same origin as the jerk when the train stopped; they are produced by noninertiality (in the given case, rotation) of the coordinate system. Naturally, the acceleration of a point caused by noninertiality of the system is absolutely real, relative to that system, in spite of the fact that there are other, inertial, systems relative to which this acceleration does not exist. In equation (8.7) this acceleration is written as if it were due to some additional forces. These forces are usually called inertial forces. In so far as the acceleration associated with them is in every way real, the discussion (which sometimes arises) about the reality of inertial forces themselves must be con¬ sidered as aimless. It is only possible to talk about the difference between the forces of inertia and the forces of interaction between bodies. But if we consider the force of Newtonian attraction, we cannot ignore the striking fact that, like the forces of inertia, it is proportional to the mass of the body. As a result of this, the equations of mechanics can be formulated in such a way that the difference between gravi- 72 MECHANICS [Part I tational forces and inertial forces does not at all appear in the equations; all these forces turn out to be physically equivalent. However, this formulation is, of course, connected with a re-evaluation and a substantial revision of the basis of mechanics. It is the subject of Einstein’s general theory of relativity, which is discussed in somewhat more detail at the end of Sec. 20. Coriolis force. Let us now consider in more detail the inertial forces appearing in (8.7), which are due to a rotating coordinate system. The first term in (8.7) occurs as a result of nonconstancy of angular velocity. It will not interest us. The second term is called the Coriolis force. For a Coriolis force to appear, the velocity of a point relative to a rotating coordinate system must have a projection, other than zero, on a plane perpendicular to the axis of rotation. This velocity projection can, in turn, be separated into two components: one, perpendicular to the radius drawn from the axis of rotation to the moving point, and the other, directed along the radius. The most interesting, as to its action, is the component of the Coriolis force due to the radial component of velocity. It is perpendicular both to the radius and to the axis of rotation. If a body moves perpen¬ dicularly to a radius, then its Coriolis acceleration is radial, and therefore analogous in its action to the centripetal acceleration which will bo considered a little further on. We note that the Coriolis force cannot be related, even formally, to tho gradient of a potential function U. There are many examples of the deflecting action of the Coriolis force in nature. The water of rivers in the Northern Hemisphere which flow in the direction of the meridian, i.e., from north to south, or from south to north, experience a deflection towards the right- hand bank (if we are looking in the direction of flow). This is why the right-hand bank of such rivers is steeper than the left. It is easy to form the corresponding component of the Coriolis force. The angular-velocity vector of the earth’s rotation is directed along the earth’s axis, “upwards” from the north pole. The waters of a river, which flows southwards at tlie mean latitudes of the Northern Hemi¬ sphere, have a velocity component perpendicular to the earth’s axis and directed away from the axis. This means that the Coriolis acceleration of the water, relative to the earth, is in a westerly direction or, relative to a river flowing southwards, to the right. If the river flows in a northerly direction, the deflection will be towards the east, i.e., again to the right. In the southern hemisphere the deflection occurs leftwards. The warm Gulf Stream which flows northwards is deflected towards the east, which is of tremendous importance for the elimate of Europe. In general, the Coriolis force considerably afiects the motion of air and water masses on the earth, though when compared in magnitude with the gravitational force it is very insignificant. Indeed, the angular Sec. 9] THE DYNAMICS OF A BIGID BODY 73 velocity of the earth, as it completes one rotation about its axis in 24 hours, is a little less than 10-* rad/sec, while the velocity of a particle of water or air can be taken as having an order of magnitude of 10® cm/sec. From this the Coriolis acceleration has an order of magnitude of 10“® cm/sec®, which is one hundred thousand times less than the acceleration caused by the force of gravity. The Coriolis force also causes the rotation of the plane of oscillation of a Foucault pendulum. With the aid of the Foucault pendulum, we can prove the rotation of the earth about its axis without astro¬ nomical observations. In a nonrotating system, the plane of oscillation must be invariable in accordance with the law of conservation of angular momentum. Centrifugal force. The third vector term in equation (8.7) is the usual centrifugal force. Indeed, it is perpendicular to the axis of rotation and, in absolute value, is equal to I m [w'[«ijr]] I = mta j [tor] | = (tor sin a) = mto®r sin a. (8.8) Here, the first equality takes account of the fact that the vectors to and [tor] are perpendicular to each other, so that the absolute value of the vector product is equal to the product of their absolute values. But r sin a is equal to the distance from the axis of rotation, so that this force satisfies the usual definition of a centrifugal force. Exercise Let cs consider the rotation of the plane of oscillation of a Foucault pendulum under the action of the earth’s rotation about its axis. The axis Ox at a given point on the earth is drawn in a northerly direction and the axis Oy in an easterly direction. Then, if <03 = « sin 0 , where 0 is the latitude of tho locality, we have the equation of motion — wja;—2yco3 , gr = — 1/-f 2a:cOg , Multiplying the first equation by y and the second by x and then sub¬ tracting, we got ^(ySo-xy)^-^{,f+x^)^^. Integrating and transforming to polar coordinates (x — rcosi y — r sin 9): Whence, after cancelling the r®’s, wo have 9 = ojg = 01 sin 0 , which gives the angular velocity of rotation of tho plane of oscillation. Sec. 9. The Bjmamics of a Rigid Body Tho d 3 mamics of a rigid body is a large independent chapter of mechanics and is very rich in technical applications. Our aim is to give only a brief account of the basic concepts of this branch 74 MBCBANICS [Part I of mechanics inasmuch as it contains instructive examples of general laws. In addition, certain mechanical quantities that characterize a rigid body are necessary for an understanding of molecular spectra. The kinetic energy of a rigid body. As was shown in Sec. 1, a rigid body has six degrees of freedom. Three of them relate to the trans¬ lational motion of the centre of mass of a body in space. The re¬ maining three degrees of freedom correspond to rotation (relative to this centre of mass). In Sec. 4, it was shown that the kinetic energy of a system consists of the kinetic energy of the motion of the whole mass of the body concentrated at the centre of mass, and the kinetic energy of the relative motion of the separate particles of the system. In the case of a rigid body, relative motion reduces to rotation with the value of angular velocity (a the same for all particles. Naturally, both the magnitude and the direction of to may vary with time. Let us calculate the kinetic energy of rotation of a rigid body. In the general case, the density p of the body may not be uniform over tlie whole volume of the body, and may depend on the co¬ ordinates : p = p (a;, y, z) = p (r). The mass of an element of volume dV is equal to dm — p dV. The velocity of rotation v is, from (8.4), [oir]. Therefore, the kinetic energy of the volume element is equal to ~ p [o>r]**dF. The kinetic energy of the whole body is represented by the integral of this quantity with respect to the volume T=lJp[«rfdF. (9.1) Expressing the square of the vector product in terms of the compo¬ nents ti>, we have [tor]® = £ 0 ® r® sin® a = to® r® — w® r® cos® a = to® r® — (tor)®. Here a is the angle between to and r. But to® = toj -f toj -1- to|, (tor)® = -f Wyy -f- to^z)® = =toJa:® -f tojy® -f to’z® -+- 2coxtOya;y -f 2t0xt0ia:z -p 2tOyto*yz. Since the body is rigid, the components to*, toy, to* can be taken out of the volume integral. Combining terms which are similar in the components to, we obtain for T: T = -i tol J p (y® -f- z®) dF -f -i to® J p (a;® -b z®) dF+ + 4"“*/P — to*tOy J pxydV — to*to* J pxzdF— — toyto* J pyzdF . (9.2) Sec. 9] THE DYNAMICS OF A RIOID BODY 76 Moments of inertia. All the integrals appearing in (9.2) depend only on the shape of the body and its density distribution, and do not de¬ pend on the motion of the body (in a coordinate system fixed in the body). We denote them as follows: Jyy=j + Z’^)dV, Jzz=^jp{x’^ + y^)dV, Jxy — J* pxy dV) Jxz— — J pxzdV, Jyz= — J pyzdv (9.3) The quantities with the same indexes are called moments of inertia, while those with different indexes are called products of inertia. In the notation of (9.3), the kinetic energy has the form T = xxiot -f" J vyCOy -|- J liiol -1- 2J xy<0x(0y "h 2JxzO^xOz “t* yzCOyOIz) . (9.4) With the aid of the summation convention used in Sec. 2, when eval¬ uating Lagrange’s equations the kinetic energy can be written in the following concise form: T — OatOp . Principal axes of inertia. Let us suppose that Oxyz is a coordinate system fixed in a body. In this system all the quantities J**, ..., Jyz are constant. Let us take another coordinate system Ox'y'z' which is also fixed in the body. The old coordinates of any point are expressed in terms of its new coordinates by the well-known formulae of analyti¬ cal geometry: x = x' cos Z. («', x) -f y' cos /_ (y', x) -f- z' cos Z (z'. x ), y = x' cos Z (*', y) + y' cos Z W, y) -f- 2' cos Z (z', y), z = x' cos z {x', z) -b y' cos Z («/', z) + z' cos Z (z', z), or, if we denote cos < (xa'x^) by the symbol then, with the aid of the summation convention a:p=a:,'Aap. The same formulae are used to express also the components of any vector, and in particular o>p, relative to the old axes, in terms of the components to*' relative to the new axes. Let us substitute these expressions into the kinetic energy (9.4) and collect the terms containing the products oix'oy', oy'o)*' and * Xi^y, a!,=*. 76 MECHANICS [Part I the squares w*'®, to/®, to^'®. We shall now show that wo can always rotate the coordinate axes so that the coefficients of the new products i-1-J 3 C 0 I). (9.5) These axes are called tlie ‘principal axes of inertia of the body ; they can be defined relative to any point connected with the body. By defi¬ nition, the products of inertia convert to zero in the principal axes of inertia. The moments of inertia in the principal axes are called princi¬ pal moments of inertia. They are denoted by Jy, J^, J3. The angular momentum of a rigid body. Let us now calculate a pro¬ jection of the angular momentum of a rigid body. From the definition of angular momentum we obtain Mx — J p[rv]*dF = J p [r[3. Comparing (9.6) and (9.4), we see that Mx = ^. (9.7) 3f y and 31x appear analogous. In vector form, we may write M = (9.8) Equations (9.7) and (9.8) again express the fact that the angular momentum is a generalized momentum related to rotation. In this sense, (9.7) corresponds to (6.4). The only difference is that the com¬ ponents u> are not total time derivatives of some quantities. This will Sec. 9] THE DYNAMICS OF A RIGID BODY 77 be shown a little later in the present section. In that sense, tax, in (9.7), is not altogether similar to 9 in (5.4). If the coordinate axes coincide with the principal axes of inertia, then the expression for angular momentum is even simpler than (9.B): <«) and similarly for the other components. Moment of forces. Let us now find equations which describe the variation of angular momentum with time. The derivative of angular momentum of a particle is “=4-M=w + ['pi=m' where the first term becomes zero since r and p are parallel. Integrating this equation over the volume of the rigid body and taking advantage of the additive property of angular momentum, we have M = J[rF]dF = K. (9.10) The right-hand side of (9.10), which we denote by K, is called the resultant moment of the forces applied to the body. If F is the gravi¬ tational force (which occurs in the majority of cases) then K can also be written as K = — J psr[rZo]dF, where Zq is the unit vector in a vertical direction. But since the vector Zf, is a constant, it should be put outside the integration sign: K = [zo, Jp^rdF]. If the body is supported at its centre of mass, then, by the definition of centre of mass, the integral for all three projections pr wifi be zero. Then K = 0 and the total angular momentum will be conserved. This occurs in the case of a gyroscope. For the conservation of angular momentum of a rigid body it is sufficient that K = 0; but for any arbitrary mechanical system, angular momentum is conserved only when there are no external forces. Euler’s equations. Equation (9.6) gives a relationship between M and CO. The quantities J XX i • • • y J yz are eonstant only in a coordinate system fixed in the rigid body itself. If we write equation (9.10) for a stationary coordinate system, then, differentiating M with respect to time, we must also find the derivatives of Jxx, ..., Jyz with respect to time, which is very inconvenient. Therefore, it is preferable to 78 MECHANICS [Part I transform the equation to a coordinate system fixed in the body, taking into account the accelerated motion of that system. The varia¬ tion of the vector M relative to the moving axes consists of two com¬ ponents ; one is due to the variation of the vector itself, while the other is due to the motion of the axes onto which it is projected. For the vector M this variation is equal to [coM], similar to the way that it was equal to [wr] for the radius vector r in Sec. 8 . When the coordinate system is rotated, any vector varies like a radius vector. Let the coordinate axes be taken in the direction of the principal axes of inertia. Obviously, the moments of inertia relative to these coordinates are constant. For this reason, the time derivative of is Ml = Ji Wj q- [<«> M]j — J-i till -f- M^ —11)3 M^ — Ji -f- ( *^3 “3 "b ('^2 ^>>1 f <>2 — -^3 • These equations were obtained by L. Euler and are named after him. They can be reduced to quadrature for any arbitrary values of integrals of motion in the following cases: 1 ) Ki = K 2 =K^ — 0 (point of support at the centre of mass) for arbitrary values of the moments of inertia; 2 ) Jjj = J 3 and the point of support lies on the axis of symmetry, relative to which two moments of inertia are equal. This is the so- called symmetrical top. For more than a hundred years, no other case of a solution of system (9.11) by quadratures was known. Only in 1887 did S. V. Kovalevskaya find another example (see G. K. Suslov, Theoretical Mechanics, Gostekhizdat, 1944). Kovalevskaya showed that the three listed cases exhaust all the possibilities of integrating the system (9.11) by quadra¬ tures for arbitrary constants (integrals) of motion. A free symmetrical top. All three cases, and in particular the Kova¬ levskaya case, are very complicated to integrate. Therefore, we shall only consider the simplified first case, when (a free symmetrical top). From the first equation of (9.11), it immediately follows that o>i=const. For brevity, we write the value —l) = Q. The second two equations of (9.11) are written thus: (9.12) Sec. 9] THE DYNAMICS OF A BIQID BODY 79 d )2 + O2 = 0. (9.13) Equations (9.13) are easily integrated if we represent the components u >2 and 0)3 in the following form; 6)2 = Wx cos Qi, 0)3 = tox sin f2<. (9.14) Here, o)| + w§ = tox is a constant quantity. Thus, the angular- momentum projection on the axis of symmetry and the sum of the squares of the angular-momentum projections on the other two axes are conserved. This means that the angular-momentum vector rotates about the axis of symmetry, i. e., the first axis of inertia, with angular velocity £ 2 ; the vector makes with it a constant angle, the tangent of which is situation m a system of moving axes. Of course, in a system of stationary axes, the total angular momen¬ tum is conserved in magnitude and direction, since the resultant moment of force is equal to zero. In this system, the axis of symmetry of the top rotates about the angular-momentum direction making a constant angle with it. Such motion is called precession. Pre- cessional motion is only stable for rela¬ tively small external perturbations. The stabilizing action of gyroscopes is based on this principle. Eulerian angles. We shall now show how to describe the rotation of a rigid body with the aid of parameters which specify its position. Such parameters are the Eulerian angles shown in Fig. 13. The figure depicts two coordinate systems: a fixed system Oan/z and a system Ox'y'z' fixed in the rigid body. It is most convenient to take x', y', z' along the principal axes of inertia through the point of support. Then the Eulerian angles are: 9^ is the angle between the axes z and z', 9 is the angle between the line OK of intersection between the planes xOy and x'Oy' and the a;'-axis, ij; is the angle between the line OK and the a;-axis. If the angle varies, then the angular-velocity vector ^ is directed along the axis Oz since that vector is perpendicular to the plane of angle of rotation <{'• Thus 9 must be taken along the axis Oz' and 9 along the line OK. Let us now express the angular-velocity projections (i. e., Wj, < 02 , 6 > 3 ,) onto the principal axes of inertia in terms of the generalized ve¬ locities <];, 9 ,9. P 80 MECHANICS [Part I < 1)3 is the projection of the angular velocity on the axis Oz' {z' is the third axis). As was shown, 9 is projected exclusively on this axis and the projection of ij; is equal to cos %■, since & is the angle between the axes Oz and Oz'. Hence, (,> 3 = 9 4 -tj^cosS-. (9.15) In order to find the projections of the angular velocity on the other two axes, we draw a lino OL which lies in the plane x'Oy' and is per¬ pendicular to OK. From Fig. 13 it can be seen that Z_L0x' = - 2 -— 9 and /_ zOL = ^ , since the straight line OL lies in the plane zz', as do all lines perpendic¬ ular to OK. The projection of ip on OL is equal to —A sin 9-, and the projection on Ox' is equal to —

= sin®9 4 ' + Pv ^ > whence The energy integral, after substituting p, and p^ is (P'j'—Pep cos a)* 2 sin® a ++ mgricos^-. (9.23) Thus, the iiroblem is reduced to motion with one degree of freedom 9, as it were. The corresponding “kinetic energy” is Ji9®, and the “potential energy” is represented by those energy terms which depend on 9. This potential energy becomes infinite for 9 = 0, and 9 =h. Hence, for 0 <9 then the rotation of the top, whoso centre of mass is above the point of support, is stable. Small oscillations are possible near the potential energy minimum. These oscillations are superimposed on the processional motion of the top which we have already noted. They are called nutations. Sec. 10. General Principles ot Mechanics In this part of the book, mechanics is explained mainly through the use of Newton’s equations ( 2 . 1 ). Going over to generalized coordinates, we obtain from them Lagrange’s equations and a series of further deductions. In this section it will be shown that the system of Lagran¬ ge’s equations can be obtained not only from Newton’s Second Law, but also from a very simple assertion about the value of the integral 6 - 0060 82 MECHANICS [Part I of tlie Lagrangian taken with respect to time. The basic laws of mechan¬ ics thus formulated are usually called integral 'principles. The particular importance of these principles is that they allow us to understand, in a unified manner, the laws relating to various areas of theoretical physics (mechanics and electrod 3 aiamics), thus opening up a field for broad generalizations. Action. For a certain mechanical system, let it be possible to define the Lagrangian L=L{q,q,t), (10.1) as dependent on the generalized coordinates q, velocities q, and the time t. We shall consider that all the coordinates and all the velocities are independent. Let us choose some continuous, but otherwise arbi¬ trary, dependence of the coordinates upon the time q (t). The functions q (t) can be in complete disagreement with the actual law of motion. The only requirement imposed on q (<) is that the functions q (t) should be smooth, i. e., that they should provide for differentiation and should correspond to the rigid constraints present in the system. The time integral of the Lagrangian is called the action of the sys¬ tem: S=jL(q,q,t)dt. (10.2) *0 The magnitude of this integral depends upon the law chosen for q (<), and is, in that sense, arbitrary. In order to examine the relation¬ ship between the action and the function q (t), it is convenient to cal¬ culate the change of S for a transition from some arbitrary law q (t) to another, infinitely close but also arbitrary, law q' (t). Yariation. Fig. 14 shows two such conceivable paths. Time is taken along the abscissa, and one of the generalized coordinates q, represent¬ ing the totality of generalized coordinates, is plotted on the ordinate axis. For the specification of future operations, we shall consider that both paths pass through the same points, q^ and at the initial and final instants of time. Fig. 14 The vertical arrow shows the difference between two conceivable, infinitely close paths at some instant of time other than initial or final. This difference is usually called the variation of q and is denoted by Sg. The symbol S should emphasize the difference between variation and the differential d; the differential is taken for the same path at various instants of time, while the variation is taken for the same instant of time between different paths. Sec. 10] GEKBRAIi PBINCITLES OF MECHANICS 83 Since the neighbouring paths in Fig. 14 have different forms, the speed of motion along them %vill also difer. Together with the variation of the coordinate 8q between paths, we can also find the variation in velocity 8g. We shall show that 8q =~8q. Indeed, 8q=q' {t)—q (t), where q' and q are values of the coordinates for neighbouring paths. d But the derivative of the difference ^ 8q is equal to the difference of the derivatives q' (f)— q(t)=8q. Let us now find the variation of the Lagrangian, i. e., the difference of the function for two adjacent paths. Since L = L {q, q, t) and the variation is taken at the same instant of time, i. e., S< = 0, we obtain 8L = dL aq (10.3) Let us rearrange the second term. Taking advantage of the fact that S q= 8q, we can write it thus: 8L _aL d aq aq at d ah dt aq The last equation simply expresses a transformation by parts. Substituting it into (10.3), we find The integral of the variation of L is equal to the variation of action 8S, since the difference between integrals taken between the same limits is equal to the difference between the integrands. The first term in (10.4) can be integrated with respect to time, because it is a total derivative. The variation of action is then reduced to the form ^0 ^0 We have agreed to consider only those paths which pass through the same points, q^ and q^, at the initial and final instants of time. Hence, at these instants the variation 8q becomes zero by convention, and the integrated term disappears. The expression SiS is reduced to the following integral: ^0 The extremal property of aetion. If the chosen path coincides with the actual path of motion, the coordinates satisfy Lagrange’s equation: 6* 84 MECHANICS [Part I d dL dL dt dq 8q (10.7) Substituting this in ( 10 . 6 ), we see that the variation of action tends to zero close to the actual path. The change in magnitude is equal to zero either close to its extreme, or close to the “stationary point” (for example, the function y = 3? has such a point at x = 0, where y’ = 0, y” =0). Three cases can, in general, be realized: a minimum, a maximum and a stationary point. For example, let a point, not subject to the action of any forces other than constraint reactions, move freely on a sphere. Then its path will be an arc of a great circle. But through any two points on the sphere there pass two arcs of a great circle representing the largest and smallest sections of the circumforence. One corresponds to a maxi¬ mum, and the other, to a minimum, 8. If the beginning and end of the path are diametrically opposite, the result is a stationary point. The principle of least action. We have proven, on the basis of Lag- range’s equations, that SS = 0. We can proceed in a different way: by asserting that close to the actual path passing between the given initial and final positions of the system the increment of action is equal to zero, we can derive Lagrange’s equations. Ordinarily, the action on an actual path is minimal, and therefore the assertion we have made is called the principle of least action. Action was wiltten in the form ( 10 . 2 ) by Hamilton. Much earlier, the prineiple of least action was mathematically formulated by Euler for the siiecial case of paths corresponding to constant energy. For us, it is not essential that the action should be a minimum, but that it should be steady, SjS = 0 . Lagrange’s equations are derived from the principle of least action by means of proving the opposite. We assume the right-hand side of equation ( 10 . 6 ) to be zero, S0. We have shown that if we proceed from the principle of least action as a requirement for the motion along an actual path, then that path must satisfy Lagrange’s equations. Sec. 10] OENERAI. PRINCIPLES OP MECHANICS 86 The advantages of using action. The principle of least action may at first sight appear artificial or, in any case, less obvious than Newton’s laws, to whose form we are accustomed. For this reason we shall try to explain where its advantages lie. First of all, let it be noted that Lagrange’s or Newton’s equations are always associated ■ with some coordinates whose choisc is, to a significant extent, arbitrary. In addition, the choice of coordinate system, relative to which the motion is described, is also arbitrary. Yet the motion of particles along actual paths in a mechanical sys¬ tem expresses a certain set of facts which cannot depend on the arbitrary manner of their description. For example, if the motion leads to a collision of particles, that fact must always be represented in any description of the system. But it is precisely the integral principle that is especially useful in a formulation of laws of motion not related to any definite choice of coordinates, the value of the integral between the given limits being independent of the choice of integration variables. The extremal property of an integral cannot be changed by the way in which it is calculated. The integral principle S;S' = 0 is equivalent, purely mathematically, to Lagrange’s equations (2.21). But in order to apply it to any actual system, we must have an explicitly expressed Lagrangian. It may be found from those physical requirements which should be imposed on an invariant law of motion that is independent of the choice of coordi¬ nate axes and the frame of reference. As a result of the invariance of the principle of least action, we can consider the laws of mechanics in a very general form, and this, therefore, opens the way for further generalizations. The determinacy of the Lagrangian. Before finding an explicit form for the Lagrangian, we must put the question; Is the determined function we are looking for single-valued ? We shall show that if we add the total time derivative of any function of coordinates and time, / {q, t), then Lagrange’s equations remain unchanged. This can be verified either by simple substitution into (10.7), or directly from the integral principle. Writing L = L'-b^/(gr,0. (10.8) we see that (10.9) 4 ^0 ^0 ^0 The variations of / appear in the variation of 8 only at the limits of integration. But since we have arranged that / depends on the coordi- 86 MECHANICS [Part I nates and time, but not on the velocities, the variation of / is expressed linearly in terms of the variations of the coordinates, and is zero at the limits of integration. Therefore, ti ) ^ ^ dt ~ dt • 88 MECHANICS [Part I However, the expression on the left-hand side of the equation can d L be a total derivative of the function of coordinates only if is independent of velocity. Introducing the notation wo obtain 8L m , "Siyij “ T “ > dt dt ’ 8L could not be put inside the derivative sign. for, otherwise, ,,, ’ ’ £)(«*) In this way we have shown that the Lagrangian for a free particle is equal to “ ' ”2 ( 10 . 12 ) The Lagrangian for a system of noninteracting particles is equal to the sum of the Lagrangians of these particles taken independently, since it is the only sum of quadratic expressions of the type (10.12) that changes by a total derivative when Vi = V/'-l-V (where i is the particle number) is substituted. In order to write down L for a system of interacting particles, we must, of course, make certain physical assumptions about the nature of the interaction. 1 ) The interaction does not dejiend on the particle velocities. This assumption is justified for gravitational and electrostatic forces, and is not justified for electromagnetic forces. It should, however, be noted that electromagnetic interactions involve ratios of particle velocities and the velocity of light c, and therefore, to the approximation of Newtonian mechanics, they must be considered as negligibly small. The Lagrangian of Neivtonian mechanics is not universal and is appli¬ cable only to a limited gi’oup of phenomena, when all Vi c. 2) The interaction docs not change the masses of the particles. 3) The interaction is invariant with respect to Galilean transforma¬ tions. From these conditions it can be seen that the interaction appears in the Lagrangian in the form of a scalar function determined only by the relative distribution of the particles: L= — (10.13) i From this expression, we can find the conservation laws for energy, linear momentum, and angular momentum (see Sec. 4). The Hamiltonian function. We shall now use the principle of least action in order to transform a system of equations of motion to other variables. Namely, in place of coordinates and velocities we shall Sec. 10] GENERAL PRINCIPLES OF MECHANICS 89 employ coordinates and momenta. Let us assume that velocities are eliminated from the relations (10.14) Since the Lagrangian depends quadratically on the velocities, equa¬ tions (10.14) are linear in the velocities and can always be solved. We shall obtain for coordinates and momenta a more symmetrical system of equations than Lagrange’s equations. The passing from velocities to momenta was performed to some ex¬ tent when we substituted the integrals of motion in the expression for energy, for example, in (5.4), (9.21), (9.22). Now, in place of the velocities we shall introduce into the energy the momenta for all the degrees of freedom, (and not only for the cyclic ones, i. e., those, whose coordinates do not appear explicitly in L). Energy expressed in terms of coordinates and momenta only is called the Hamiltonian function of the system or, for short, the Hamiltonian-. ^ [?>? iP)] = i 0 or the jirojection of rot A, normal to the area at the given point, is the limit of the ratio of the circulation of A, over the contour of the area, to its value when the contour is contracted into the point. So that the integral JA d\ should not become zero, wo must have closed vector lines, to some extent following the integration contour, which lines are similar to the closed lines of flow in a liquid during vortex motion. Hence the term curl, or rotation. If the circulation is calculated from a finite contour then the contour can be broken up into infinitely small cells to form a grid. For the sides of adjacent cells, the circulations mutually cancel since each side is traversed twice in opposite directions; oidy the circulation along the external contour itself remains. The integral on the right- hand side of equation (11.17) gives the flux of rot A across the surface “pulled over” the contour. Thus, we obtain the desired integral theorem |'Adl=JrotAds, (11.19) which is called Stokes’ theorem. Differentiation along a radius vector. The divergence and rotation of a vector are its derivatives with respect to the vector argument. They can be reduced to a unified notation by means of the following. We introduce the vector symbol V (nabla*) with components V.v = a bx ’ V. a bz Then, from (11.7), we obtain for the divergence of A : ( 11 . 20 ) * Nabla is an ancient musical instrument of triangular shape. This symbol is also called del. 7 - 00«0 08 ELECTKOD YNAMICS [Part II div Vx^*+Vy^v+V^^^ = {V^), ( 11 . 21 ) i.e., a scalar product of nabla and A. From (11.16), we have for the rotation rot A r-i i (Vy Ai .— Vi A y) -j- j (Vi A* — V* Ai) -|- k (V* Ay — Vy A*) ■— sa[VA]. (11.22) We use the identity symbol -. here in order to emphasize the fact that we are simply dealing with a new system of notation. We shall see, however, that this system is very convenient in vector analysis. We note, with reference to algebraic operations, that nabla is in all cases similar to a conventional vector. We shall use the expressio)\ ‘‘multiplication by nabla” if, when nabla operates on any exjiression, that expression is differentiated. Sometimes, nabla is multi})lied by a vector without operating on it as a derivative. In that case it is ai^idied to another vector [see (11.30), (11.32)]. Gradient. If we operate with V on a scalar cp, we obtain a vector which is called the gradieivt of the scalar cp: grad9^V9 = i||+j + (11.23) Its components are: (11.24) From ecjuations (11.24), it can be seen that the vector Vcp is per¬ pendicular to the surface

• Sec. 11] VBCTOa ANALYSIS 103 2) Write down A']; in cylindrical coordinates. 3) AVrite down the three components of AA in spherical coordinates. 4) Two closed contours are given. The radius vector of points of the first contoiu" is Ti, of the second contour, r^. The elements of length along each contour are cilj and d\^, respectively. Provo that the integral is equal to 'zero, 4 n, 8 w, and 9 i -4 it, depending upon how many times the first contoiu’ is wound round the second, linking up with the latter. Vj denotes diffe¬ rentiation with respect to Tj (Ampere’s theorem). Changing the order of integration and performing a cyclic permutation of tho factors, we have 1 dl ■])• We apply Stokes’ theorem (11.19) to the integral in dli: 1 roti Vi d\ s] ci'Si) • •l— *2 I Wo use equation (11.30); rotj denotes dilTerentiation with rospoct to tho compo¬ nents Fi; and dl^ in such a differentiation may be regarded as a constant vec¬ tor: dlj = (dljVdVi- -dlj divtVi- roti Vi In accordance with exercise la, the last term containing Aj to zero. There remains, therefore, 1 ivi-iv— a: 1 Ti—'•2r is equal roti dl, i] = (dljVi) Vi- - — (<^4 V 2 ) Vi ■ since a function of the difference Fj—Fj is differentiated. For short, we write f^Fj—Fj. Then the required integral will be « = — J (dU V2) J dSi Vi I = — / (<^*2 V 2 ) J V ^ . We shall now explain tho geometrical sense of the second integrand, i.e., , 1 (dStl) 1 rm 1 J . (f^slr) . „ dSi V — =-^ • Tlie scalar product —-—- is the projection of an element of the surface dSj pulled over the first contour, on the radius vector r drawn from a point on the second contour. In other words, is equal to the projection of tho area dSj on a plane perpendicular to r. This projection, divided by r“, is equal to the solid angle dSi at a point Fj on the second contoiu- C {rf 8 1 f*) subtended by the area ds^. The integral - ' is, therefore, that solid angle a which is obteuned if a cone is drawn with vertex at the point r^, so that the generating line of the cone formed the contour Ij. The differential (dlj Vj) Cl is the increment of solid angle Cl obtained in shifting along the contour Zj a distance dij. Thus, a = J (dljV 2 ) ^ = f^i- -n. The integral of this quantity around a closed contour is equal to the total change in solid angle in traversing the contour Zj. Let the initial point of cir¬ cumvention lie on the surface Sj. Then the solid angle subtended by the surface 104 KLKCTUODYKAMICS [Part 11 at the origin is — 2 7t. If the contours arc linked, then the solitl angle will bo 2 -K after the circumvention, since the area i.s observed from a terminal point on the other side. If the contours are not linked, then the solid angle is once again its initial value, — 2 re, and the integral is equal to zero. Thus, when the contours are linked n times, the integral in lil.^ is equal to 4 re «. See. 12. The Eleetromagnelic Field. Maxwell’s Equations Interaction in mechanics and in electrodynamics. The interaction of charged bodies in electrodynamics is principally an interaction of charges with an elcctronittgiietic licld. However, the jthysical concept of the field in electrodynamics differs es.sentially from the field concept in Newtonian meehtinics. We know that the s]tace in which gravitational forces act is called a gravitational licld. The values of these foi ces at any point of tho field is determined, in Newtonian mechanics, hy the instantaneons positions (ff the, gravitating bodies, no matter how far they are from the given ])oint. In electrodynamics, such a field representation is not satis¬ factory : during the time that it takes an electromagnetic disturbam-e to move from one charge to another, the latter can move a very great distance. IClcmentary ch.arges (electrons, protons, mesons) veiy often have velocities close to the velocity of pro])agation of electromagnetic disturbances. Modem gravitational theory (the general theory of relativity', sec Kec. 20) shows that gravitational interaction, too, prop.agates with a finite velocity. But since macroscopic bodies move considei'ahly slower, within the scale of the solar sy^stem, the finite v'eloeity of pro])agation of gravitational forces introduces only' an in.significant correction to the laAvs of motion of Newtonian mechanics. In the elcctrodymamics of elementary' charges, the finite velocity of ])ro])agation of electromagnetic disturbances is of fundamental signi- licanco. When si)caking of point charges, the action of a field on the charge is always determined only by' the field at the point where the charge is located, and only' at the instant when the charge is at this point. As opposed to the “action at a distance” of Newtonian mecha¬ nics. such interactions arc termed “short-range.” If the energy or momentum of a charged particle is changed under the action of a field, they' can he imparted directly only to the electro¬ magnetic field, since a finite interval of time is necessary for the energy and momentum of other particles to be changed. But this means that the electromagnetic field itself possesses energy' and momentum, whereas in Newtonian mechanics it was sufficient to assume that only the interacting particles possessed energy and momentum. It follows from this that the electromagnetic field is itself a real physical entity to exactly the same extent as tlie charged particles. The equations of electrodynamics must describe directly the propagation of electro- See. 12] THE ELECTRt)MAGNETIC FIELD. MAXWELL’S EQUATIONS 105 magnetic disturbances in space and the interaction of charges with the field. Interaction between charges is effected through the electromagnetic field. Such laws as the Coulomb or Biot-Savart laws (in which only the instantaneous positions and the instantaneous velocities of the charges appear) are of an approximate nature and are valid only when the relative velocities of the charges are small compared with the propagation velocity of electromagnetic disturbances. It will be shown later that this velocity is a fundamental constant which appears in the equations of electrodynamics. It is equal to the velocity of light in vacuo and, to a high degree of precision, is 3 X 10*® cm/sec. A field in the absence of charges. The independent reality of the electromagnetic field is particularly evident from the fact that electro¬ dynamic equations admit of a solution in the absence of charges. These solutions describe electromagnetic waves, in particular light waves, in free space. Thus, electrodynamics has shown that light is electromagnetic in nature. In the course of two centuries, the protagonists of the wave theory of light considered that light waves were propagated by a s^iecial elastic medium permeating all space, the so-called “ether.” In order to represent the .spread of oscillations it was, naturally, necessary to have something oscillating. This “something” was called the ether. Proceeding from an analogy with the propagation of sound waves in a continuous medium, the ether was endowed with the ]noperties of a fluid, physical phenomena being explained simply by reducing them to definite mechanical displacements of bodies. In particular, light ])hcnomena were regarded as displacements of particles of the special medium, the ether. In this, a peculiar “abhorrence of a vacuum” was apparent or, more exactly, a purely speculative representation of empty space where “nothing exists” and, hence, where nothing can occur. Physicists did not at once come to realize that the electromagnetic field itself was just as real as the more tangible “ponderable matter.” Electrodynamic laws are those elementary concepts, from which the interaction of atoms should be deduced, which interaction accounts for the proper¬ ties of real fluids that are incomparably more complicated than the properties of a field in “empty space,” i.e., in the absence of charges. There is no sense in reducing a field to an imaginary fluid merely in order to avoid the idea of “empty space.” Physical space is the carrier of the electromagnetic field and is, therefore, inse]iarable from the state and motion of real objects. As regards the term “ether,” which still persists in the field of radio, it expresses nothing other than the electromagnetic field. The electromagnetic field. liCt us now establish the basic equations of electrodjmamics. We shall proceed from certain elementary laws. 106 ELECTRODYNAMICS [Part II which Ave assume tlie reader knows from a general course of physics or electricity. These laws will first be used in the ab.sence of matter consisting of atoms or, as is usually said in elcctrodjmamies, in the absence of a “material medium.” By this term wo must not under¬ stand any encroachment on the material nature of the electromagnetic field itself. From the elcctrodynamic equations for free space we shall, later on, derive the c(iuations for an electromagnetic field in a medium (a comlnctor or dielectric). As is known, the electromagnetic field in a medium is described by four vector /cm'/» • sec. If we substitute charge, expressed in this system of units, into the equation for Coulomb’s laAV, then the interaction force between charges is expressed in dynes (gm • cm/sec ^). Electromotive force. Let us recall the definition for electromotive force in a circuit: this is the work performed by the forces of the electric field Avhen unit charge is taken along the gteen closed circuit. And it is absolutely immaterial what the given circuit represents: whether it is filled Avith a conductor or Avhether it is merely a clo.sed line drawn in space. Let us Avrite doAAui the expression for electromotive force (abbreviated as e. m. f.) in the notation of Sec. 11. The force acting on unit charge at a given pouit is the electric field E. The work done by this force on an element of path dl is the scalar product Edl. Then, the Avork done on the Avhole closed circuit, or the e. m. f., is equal to the integral e.m.f. = j Edl. (12.1) Sec. 12] THE ELECTROMAGNETIC FIELD. MAXWELL’S EQUATIONS 107 Magnetic-field fiux across a surface. Let us suppose that some surface is bounded by the given circuit. We shall denote the magnetic field by the letter H. The magnetic-field flux through an element of the chosen surface is, by the definition given in Sec. 11, d 4> = II(i!8. The magnetic-field flux through the whole surface, bounded by the circuit, is 0=)'Hds. (12.2) It can be conveniently represented thus. Let us consider a section of the surface through which unit flux A O = 1 (in the CGSE system) jiasses. We draw through this section of the surface a line tangential to the direction of the field at some point on the surface. A line which is tangential to the direction of the field at its points is called a magnetic line of force. For this reason, the tot.al fiux is equal, by definition, to the number of magnetic lines of force crossing the surface. Magnetic lines of force are either closed or extended to infinity. Indeed, a magnetic line of force may begin or end only at a single charge, but separate magnetic charges do not exist in nature. In a permanent magnet the lines of force are completed inside the magnet. From this it follows that a magnetic fiux through any surface, bounded by a circuit, is the same at a given uistant. Otherwise, a number of the magnetic Imes of force would have to begin or end in the space between the surfaces through which different fluxes pass. Consequently, at a given instant, a eonstant number of magnetic lines of force, i.e., a constant magnetic field flux passes across any surface bounded by the circuit. Therefore, the flux can be ascribed to the circuit itself, irrespective of the surface for which it is calculated. Faraday’s induction law. Faraday’s induction law is written in the form of the following equation: —( 12 . 3 ) If all the quantities are expressed in the CGSE system, then the constant of proportionality c is a universal constant with the di¬ mensions of velocity equal to 3 X 10^° cm/sec. Usually, Faraday’s law is applied to circuits of conductors; however, e. m. f. is simply the quantity of work performed by unit charge in movmg along the circuit, and, for a given field value through the circuit, cannot depend upon the form of the circuit. The e. m. f. is simply equal to the integral J Edl. In a conducting circuit, this work can be dissipated in the generation of Joule heat (“an ohmic load”). However, it is completely justifiable to consider the circuit in a vacuum also. In this case, the work performed on the charge is spent in increas¬ ing the kinetic energy of the charged particle, as, for instance in the case in an induction accelerator, the betatron. K-K pLEOTKODYNAMIOS [Part II Maxwell’s equation' Joy rot E. Thus, equation (12.3) refers to any arbitrary closed circuit. We suljstitute the definitions (12.1) and (12.2) into this equation: (' 2 ‘) The left-hand side of the equation can he transformed by the Stokes theorem (11.19) and, on the right-liand side, the order of the time differentiation and surface integration can be interchanged, since they are performed for independent variables. In addition, taking this integral over to the left-hand side, we obtain J(rotE4-l^l)ds = 0. (12.5) But, the initial circuit is completely arbitrary, i.c., it can liave arbitrary magnitude and sliape. Let us assume that the integrand, in ])arentheses, of (12.5) is not equal to zero. Then we can choose the sur¬ face and the circuit that bounds it so that the integral (12.5) does not become zero. Thus, in all cases, tlie following equation must be satis¬ fied : rot E-f- 0. *'’ (12.5) Tn comparison with (12.3), this equation does not contain anything new ])hysically; it is the same induction laAv, but rewritten in differen¬ tia) form for an infinitely small circuit (contour). In many applications the differential form is more convenient than the integral form. We shall see later that the constant c is equal to the velocity of light in free space. ■'' The equation for 4iv II. As wo have already said, magnetic lines of force are either closed or go off to infinity. Hence, in any closed surface, the same number of magnetic-field lines enter as leave. The magnetic- field flux in free space, across any closed surface, is equal to zero: j'llds=0. (12.7) Transforming this integral to a volume integral according to the (lauss-Ostrogradsky theorem (11.6), wo obtain ('divIIdF=0. (12.8) Due to the fact that the surface bounding the volume is completely arbitrary, we can always choose this volume to be so small that the integral is taken over the region in which div H has constant sign if it is not equal to zei*o. But then, in spite of (12.7) and (12.8), j div H d F will not be equal to zero. For this reason, the divergence of H must become zero: Sec. 12J THE ELECTROMAGNETIC FIELD. MAXWELL'S EQD.ATI0N8 109 divH = 0. (12.9) (12.9) is the differential form of (12.7) for an infinitely small volume. In Sec. 11 it was shown that the divergence of a vector is the density of sources of a vector field. The sources of the field are free charges from which the vector (force) magnetic-field lines originate. Thus, (12.9) indicates the absence of free magnetic charges. Equations (12.6) and (12.9) are together called the first pair of Maxwell’s equations. . o Let us now mtrqduce the^ second pair. The equation for div E. The electric-field flu.x through a closed surface is not equal to zero, but to the total electric cliarge e inside the surface multiplied by 4 tc (Gauss’ theorem); ' Ji .1 r jEds — 4n:e. (12.10) This theorem is derived from Coulomb’s law for point charges. The field due to a iioint charge e is expressed by tlie following equation: Here, r is a radius vector drawn from the point situated at the charge to the point where the field is defined. The field is inversely projior- tional to r® and is directed along the radius vector. Let us surround the charge by a spherical surface centred on tlie charge. The element of surface for the sphere ds is r^dfl where r ^ (l Q is an elementary solid angle and -j- indicates the direction of the normal to the surface. The flux of the field across the surface element is Eds-4'--»'*(^ii- = edQ. r r The flux across the whole surface of the sphere is J edO = eJ dQ — = 4 TV e. But since lines of force begin only at a charge, the flux will be the same through the sphere as through any closed surface around the charge. Therefore, if there is an arbitrary charge distribution e inside a closed surface, then equation (12.10) holds. In order to rewrite this equation in differential form, we introduce the concept of charge density. The charge density p is the charge con- tained in unit volume, so that tke total diarge in tne volume is related to the density by the following equation: e = JpdF. (12.11) A 6 Hence, p = lim . Introducing the charge density in (12.10), wo obtain 110 ELECTUOD y NAIIICS [Part II |'( the preceding sectioq by proceeding from certain simple physical laws and the assumption about the magnetic effect due to displace¬ ment current. In this section, Maxwell’s equations will be reduced to the variational principle, which is the principle of least action for the electromagnetic field. 1 Electrodynamics is not eqiTWalent to the mechanics of particle systems or to the mechanics of liquids, which are based on Newton’s laws. All the same, to a very coiLsidcrable extent, electrodynamical laws are analogous to the laws of mechanics. This analogy can best be seen from the principle of least action for the electromagnetic field. The variational formulation best of all allows us to derive the conservation laws for the electromagnetic field. The corresponding integrals of motion for a field coincide with the well-known mechani¬ cal integrals^—energy, linear momentum, and angular momentum. In a closed system consisting of charged particles and a field, the total energy, total linear momentum, and total angular momentum of the charges and field arc conserved. In this sense, electrodynamics is indeed “a dynamics” of the elec¬ tromagnetic field, though this by no means signifies that the laws of electrodynamics can be obtained from Newton’s laws. Both are equivalent to certain integral variational principles, but the action functions arc, of course, of entirely different form. It is a noteworthy fact that Maxwell at first tried to construct mechanical models of the ether, but in his later work he rejected them and obtained the general equations of electrodynamics by means of a generalization of known elementary laws of electro¬ magnetism. The Lagrangian function for a field. In order to formulate the prin¬ ciple of least action it is necessary to have an ex{)re.ssion for the Lagrangian. The choice of Lagrangian in mechanics is determined by considerations based on the relativity principle of Newtonian mechanics, which is formulated with the aid of Galilean transfor¬ mations (Sec. 8). As will be explained in detail in Secs. 20 and 21, Galilean transformations are not valid in electrodynamics and are replaced by the more general Lorentz transformations, based on the Einstein relativity principle. These transformations allow the Lagrangian for the electromagnetic field to be uniquely found; this will be done in Sec. 21. In this section, the choice of Lagrangian is justified by the fact that the already familiar Maxwell equations 118 B LBCTRO B YNAMICS [Part II are obtained from it. Similarly, in Part 1, the principle of least action was formulated after Lagrange’s equations had been obtained on the basis of Newton’s laws. This confirmed the truth of the integral principle. In finding the Lagraiigian for a system of free particles, a summa¬ tion is performed over the coordinates of the particles. The electro¬ magnetic field, if we use the terminology of mechanics, is a system with an infinit4-P9'-/(divj+ 1^)1. (13.10) However, the integrated terms do not atfect the Maxwell equations since, when performing a variation of /Sj, both 5^ L and Sa L are equal to zero at the boundaries of the integration region. We liave already encountered this in ( 10 . 0 ). The term, jiroportional to /, under the integral sign, is multiplied by the quantity div j + ’ which is identically equal to zero according to the (Oiarge conser¬ vation law (12.18). Thus. “ or R r is satisfied. We shall consider the limiting case, when v c. Then the region of applicability of our approximation will bo very large. Equations (14.1), (14.2) are called the equations of electrostatics, and (14.3) and (14.4), the equations of magneto.statics. Scalar potential in electrostatics. In order to satisfy equation (14.2), we put E = .—V 9 . (14.5) According to (14.29), 9 is the scalar potential. The equation for the scalar potential is obtaining from (14.1) div grad 9 == A 9 = — 47 tp , (14:.6) which also follows from (12.38) if we equate to zero the noastatic term -T-f. Let us find the solution to equation (14.6) for a point charge, i.e., we put p equal to zero everywhere except at one point of space. Let us put the origin at this point. Then 9 can depend only on the distance from the origm r. In Sec. 11 an expression for the Laplacian A was obtained in spherical coordinates (11.46). In the special case, when the required function depends only on r, we obtain from (11.46) 126 ELECTKODYNAMICS [Part II _1 r»' dr (14.7) Let us integrate this equation between and r^, first multiplying it by r^. Since the region of integration does not contain the origin, where the point charge is situated, the integral of the right-hand side becomes zero. Hence, rl df dr. = rf = const. Therefore the potential is 9 = —~ + i?. T j. Tlio constant B is equal to zero if we take the potential to be eqxial to zero at an infinite distance away from the charge. Let us now determine the constant A. For this, we integrate equation (14.6) over a certain sphere surrounding the origin. Since the Laplacian A 9 is div grad 9 , the volume integral can be transformed into an integral over the surface of the spliere. This integral is |^grad9ds---J r‘^clQ.= . On tlie right-hand side we have — j •iiz^dV — — 47 te, since the integration region includes the point where the charge is situated. Thus A -- — e. The potential of tlie point charge is 9-',. (14.8) We obtain the same thiixg for a spherically symmetrical volume- charge distribution, if the potential is calculated outside the volume occupied by the charge. In other words, the potential of a charged sphere at all external points is the same as the potential of an equal point charge situated at the centre of the sphere. A similar result is, of course, obtained for the gravitational potential. This fact is used in most astronomical problems, where celestial bodies are con¬ sidered as gravitating points. If the origin does not coincide with the charge, and the charge coordinates are x, y, z (radius vector r) then the potential at point X, Y, Z (radius vector R) is _ e _ e I » — *■ 1 “ V' ■(■yUjc)* + (r — y)»”+ (14.9) Sec. 141 THE ELECTUOSTATICS OP POINT CHAKGES 127 The potential of a system of charges. The potential due to several charges Cj, Cg, 63 , a, ..., whose positions are given by the radius vectors r^, r^, ..., r', at the point R. is 9 = y .. = y - - - -"w_^ - -. y |K —r'l ~ y‘)^ + {Z — z‘)^ Using the summation convention, any radicand in this formula can be rewTittcn in the form (Xx— (Xx—xl). But, in order to save space, we sliall use the notation {Xx — a:[) ^ instead of (Xx — a:j^) (Xx — x^) . Then the potential due to a system of point charges is written as I But we must remember that inside the brackets is a summation for X from 1 to 3. Note also tliat the potentials due to separate chai’gcs at the point R are additive, since equation (14.6) is linear in 9 . And so the full solution, due to all the charges, is equal to the sum of all the partial solutions for each charge separately. The potential due to a charge system at a large distance. Let us now assume that the origin is situat<>d somewhere inside the region occuiiied by the charges (for examjile, at the centre of the smallest sphere embracing all charges), and that all the radius vectors r‘ satisfy the inequalities |R|>|r'’|. (14.11) In other words, we shall look for the potential of a system of charges at a great distance from it. Then the function (14.10) can be expanded in a Taylor’s series in terms of y‘, z‘. We shall perform the exiiansion up to the quadratic term inclusively, but we shall first write it oidy for one term of the sum over all the charges, omitting the index i. The expansion is of the form: [(X -x)^+iY-yr + (Z- 2)2]-'/. = [(Xx - xx)T'l‘ = = [xa-v. J [xa-''- + ~v. • <' *• > 2 ) The summation convention permits writing in concise form the Taylor series for a function of several variables. Since XI — R^, we obtain the expression for the first derivative & rxr2i_I/_ ^ 1 d li S 1 JCil 1^^1 ’ “ Jx^ ~ ax7 Jlf Ti ^ ■ (14.13) 128 Kr.IXJTUOlJ VN'AMICS [Part II where Ave iiave used equation (11.36) which, in the notation of this section, is of tiie form . oAn li Tims, the term in tlu; .sum (14.12), wliich is linear in x^, i.s equal to I //) -!-cZ _ rR (14.14) It is .somewhat more difficult to calculate the term which is (juadratic in x~j. VVe first write down the .second derivative: 'C'^ f 0 X(t _ I 0A'|i y d 1 0AV0A'v /e v.w /,'■> SA'., dX; ' 'fhe partial derivative is equal to zero for a /- v and to 1 for <) Jvv * [X — V. Further, b 1 bR 0__1 _ A'v 3 3.Vv 0 A'v 'A‘'‘ 0A'v bR~l¥ If ■ ' ■ ' “/?”» hy the rule for differentiation of involved functions. Thus, wo obtain 0* I SA'^ A\ 77 1 _ 0 AV /7» 0A'v R-> Hem^e, the required e.vpansion | R — r | is of the form I l iR--rl ' n rll i_ 3A',xXv_ 1 dX^\ (14.15) VVe shall iiow subtract from the (quadratic term a quantity equal to zero: r’* I f - I _ .r^ I . 1 3 ca; R ' 6 ^ ft 0 from (14.iS)]. Then the last term in (14.15), written in terms 0 A' 0 T _ 0 A‘ 8 Y [a;,- of com])onents, is (taking advantage of the fact that Yy- ^ bZ -i, i'X bX c r = 0) <>z <’ y ' bZ < z :tx- I \ , 9 1 ' 3 !••= 1 R" ' ■ R^ I H- ( R^ " ft“ + / :iZ’- 1 \ , <) I3XY\ \ 77» ft3 i + I- 3 (A'2 + -I- RH ft>/ • Here, it is quite obvious that a term equal to zero has been sub¬ tracted, for X“-i-Y^-i-Z^ = }2^. UeaiTanging the terms, we have -vJ.) + (^*-4)(^-iir) + (.=-4)- ]■ \ ft* ft» I + ftS , axz , , 3rz -•^’-ft“ + “ 2 '^-ft»" Sec. 14] THK ELECTROSTATICS OF POINT CHARGES 129 The expansion (14.15) must be substituted in the equation for potential (14.10) and summed over all the charges. We introduce the following abbreviated notation; 2^ e,r‘-, I q=cx= 2^ - - i qyy = 2! “ i ^1\ 3 /’ 3 /’ '■'* \ . 3') ’ q^y=- 2J^‘^‘y‘’ i qxz --= 27 > I qy^ = 27 y‘ • (14.16) (14.17) (14.18) The vector d (i.e., the three quantities dx, dy, dz) and the six quantities qxx, qyy, qzz, qxy, qxz, qyz, depend only on the charge distribution in the system, and not on the place at which the potential is determined. In the notation of (14.16)-(14.18), the potential at large distances away from the system is of the form i;6,- , (dR) ,1 / : 'p - ■ iT + ~R^~ + r 3X^tXv Ro dXyJ' (14.19) with the terms of different indices of the type qxy actually appearing twice in the summation (for example, q^^ and the equal term q^i). The vector d is called the dipole moment of the charge system. The six quantities q are called the quadrupole moment comiionents. The dipole moment. We shall now examine the expression obtained for potential. The zero term corresponds to the approximation according to which all the charge is considered to be concentrated at the origin. In other words, it corresponds to a substitution of the entire charge system by a single point charge. This approximation is clearly insufficient when the system is neutral, i.e., if 27«. = o- I This case is very usual, since atoms and molecules are neutral (their electronic charge balances the charge on the nuclei). 9 - 00«0 130 BI^CmiOD YNAMICS [Part II Let us assume that the total charge is equal to zero and then consider the first term of the expansion, involving the dipole moment. This term decreases like^, i.e., more rapidly than the potential of the charged system. Besides, it is proportional to the cosine of the angle between d and R. The simplest thing is to produce a neutral system by taking two equal and opposite charges. Such a system is called a dipole. Its moment is d = 27« The relations in the second column are obvious. In Sec. 9 it was shown that moments of inertia can be reduced to principal axes, i.e., a coordinate system can be found for which the products of inertia are zero. But since the relations between q and J are true for any coordinate system, the components of the quadrupole moment of different signs also become zero in these same principal axes. In the principal axes, the quadrupole moment is expressed, in terms of moments of inertia, as 32 ~ ( y^Cjr^, i.e.,it indicates I* f a charge distribution extending along the z-axis. From (14.19), the potential due to such a quadrupole with one component q is 1 - _ J_\ \ R^l 4 J{^ R” I ' 2 \ R" ft’ ) 3 1 - T'?l IX^-- C ys - 2Z^\ ' R- 3Z2 \ , " 7 - - -4 J5-(1 Scos^H). (14.24) Thus, the potential of a quadrupole depends on the angle 9- according to the law 1 - 3 cos^ 9^, where D- is the angle between the axis of symmetry of the quadrupole and the radius vector of the point at which the potential is determined. Similar deviations from spherical symmetry have been found in the electrostatic potential of many nuclei. The quadrupole moments of nuclei give us an insight into their structure. The energy of a system of charges in an eleetrostatie field. We shall now calculate the energy of a system of charges in an external electric field. The potential energy of a charge in a field is equal to f/ p 9 , because the force acting on the charge is equal to F - VC/- — eVcp = eE. The energy of a system of charges is thus U^ye,rf(r'), (14.25) where r' is the radius vector for the ith charge. Let us suppose that the field does not change much over the space occupied by the charges, so that the potential at the site of the ith charge can be expanded in a Taylor’s series: cp (r) = 9 (0) + x, (- 1 ^)+ IX, X, (14.26) 134 BliBCTBOD YN Allies [Part II We shall transform the last term in the same way as in the expansion (14.16); taking advantage of the fact that

' bxdy 9 ( 0 ) 2 ^ e,- + 27 ; 8*9 dxdz 8*9 I ■ (dEo) + -o 31*:’ dxtjL 8xv (14.27) Here, the value of the field (V 9 )o = Eo at the origin has been substituted into the term involving the dipole moment. Relating equation (14.27) to the jirincipal axes of the quadrupole moment, we get 17= 9 (0)2Je. - (dEo) - ( 7 , ^ + 32 + 32 - (14-28) In the case of a neutral system, the term involving dipole moment is especially important. The quadrupole terra accounts for the ex¬ tension of the system, since it involves field derivatives. It is interesting to note that if the system is spherically symmetrical, i.e., if it has a quadrupole moment equal to zero, there is no correction to finite dimensions. Higher order corrections are also absent, so that the potential energy will always depend only on the value of the potential at the centre. This is why spherical bodies not only attract, but are also attracted, ai6) i i since r‘=:V‘. The zero term of the expansion is a total derivative and, by the steady-state condition, is equal to zero. We now transform the first term of the expansion, using the identity 0= *■' =27^- +i7«. (»v0r'. (15.16) 1 1 i From this identity it follows that in (15.15) we can substitute half the difference of the expressions on the right-hand side of (15.16). Then the vector potential will be ^ ~ = - -Oal-ZcaRLi-'VJJ. (15.17) I We now interchange the signs of the summation and vector product: A = R e,- [r*' t‘] ^ 2f (15.18) Magnetic moment. The sum appearing inside the brackets in (15.18) is called the magnetic moment of the system of charges (or system of currents). The mean magnetic moment is written thus: 1^ = 27 (15.19) i The equation for the vector potential (15.18) can be written, by means of the magnetic moment, in the following form: The field of a magnetic dipole. Let us now calculate the magnetic field. By definition H = rot A = rot , jij . Since p! is a constant vector, equation (11.30) gives H = (P V) V- P A- (pi V) . Sec. 16] THE MAOITETOSTATICS OF POINT CHABOES 130 because A“^ =0- Further, (j* V) R=n [see (11.36)]. In order to cal¬ culate we use equation (11.36). This yields (ji V) ^ = (fi V -^) == - (tl V12) = ie‘ Finally, collecting both terms, we arrive at an equation for H; „ _ 3B(BiI)— (16.21) For comparison, we deduce the expression for the electric field of a dipole: E Vm- rr (»d) _■ 3B(Bd) —B»d (16.22) Thus, both expressions for the field (both electric and magnetic) are of entirely analogous form. The only difi'crence is that, instead of the electric moment, the equation for magnetic field involves the magnetic moment. This explains its name. In the case of a charge moving in a flat closed orbit, the definition of magnetic moment (15.19) coincides with the elementary definition of moment in terms of “magnetic shell.” As was shown in Sec. 6 [see (5.2), (5.4)], the product [rv] is twice the area swept out by the (is radius vector of the charge in unit time, or [rv] = 2 . By definition of the mean value (15.4) 1 * I r ® to J c dt cto o (1.5.23) Here, is the time of orbital revolution of the charge. In this time, the charge passes every point on the orbit once; hence the mean current is equal to I—-^. This yields the definition of magnetic moment familiar from general physics: (16.24) The similarity of equations (16.21) and (15.22) shows the equivalence of a closed current (i.e., a magnetic shell) and a fictitious dipole with the same moment p,. The field at large distances from a system of currents is produced, as it were, by the effective dipole. The relationship between magnetic and mechanical moments. An especially interesting case is that when all the charges of a system are of the same kind (for example, when they are all electrons). 140 ELECTBOD YNAMICS [Part II Then the magnetic moment is proportional to the mechanical moment. Indeed, for a system of charges with identical ratios —, we obtain 1 ^ -2T E--^.E t'f'p’i ^ 'Imc M i I i (15.25) Equation (15.25) has very important applications. A system of point charges in an external magnetic field. We now consider the question of the interaction of a system of currents with an external magnetic field. For this we must have an equation describing the interaction of a point charge with the field. We obtained equation (13.17) for the general spatial charge distribution. In this equation, the transition to point charges is obtained by changing the integral to a summation over the charges. The term obtained for action is of the form ^ J - e.- 9.), (15.26) i where the indices i in A and 9 denote that the potentials are taken at the same point as the ith charge. In magnetostatics, only slowly moving charges for which v<^r are studied. Newtonian mechanics can then be applied to their motion. In the absence of a field, the action function of the particles is of the form It will be shown in Sec. 21 that this expression holds only when v-4 c. In magnetostatics, where this condition is satisfied, the action function of a system of charges in an external field is obtained by adding (15.27) and (15.26): .S' - - !■ S. - ftf). (15.28) The field due to the charges themselves does not appear in this equation. The expression for the integrand is the Lagraugian of the system. It involves velocity linearly as well as quadratically (iii the expression for kinetic energy), and, for this reason, does not have the form that we used in Part I, L—T — U. However, the general relationships still hold. Therefore, from the Lagrangian the expression for momentum is obtained in terms of velocity dr mi V' •A... C (15.29) Sec. 15] THE MAGNETOSTATICS OF POINT CHARGES 141 Let US determine the energy in terms of momentum using the basic equation (4.4): 27 Vpi - L = 2^ (w(, A. v' - —2— - -®'- A. V' + ei9i) - = 2 —2— + ei9i . (15.30) SO that the term which is linear with respect to velocity is eliminated from the expression for energy in terms of velocity. The Hamiltonian function for a system of charges in an external magnetic field. The linear term in velocity of the Lagrangian alfects the form of the Hamiltonian. Let us write down Jf" from its definition (10.15). To do this it is necessary to substitute into the energy ex¬ pression, by means of equation (15.29), momenta instead of velocities: The Hamiltonian is (15.31) (15.32) Let us assume that the magnetic field, in which the system is situated, is weak and uniform (at least within the limits of the system). The vector potential for a homogeneous field will be represented as A:.. --[Hr]. (15.33) Indeed, then rot A = H [from (11.30), (11.35) and (11.33)]. And also divA = 0 from (11.29) and (11.34). Since the magnetic field is weak we can neglect in (15.32) the term involving Af. Then, substituting (15.33) in (15.32), we find an ex- ])ression for Jf': I The last term in (15.34) gives the required addition to the Hamiltonian. Since this term is proportional to H, we can replace pi by 7«, v‘ in it to the same accuracy, i.e., neglecting terms of order H^. Performing a cyclic permutation of the factors in (15.34) and putting the sum inside a vector product sign, we obtain an expression for the addition to the Hamiltonian: .T' = - (h [r'Vj) = - (H^i). (15.36) t This expression is very similar to the energy of a system of charges in a homogeneous electric field which involves only the electric dipole moment of the system of charges. Note that this term is of the form ELECTROD VNAMICS 142 [Part II —(dE) [see (14.28)]. This indicates a further similarity between electric and magnetic moments. Larmor’g theorem. Let us now compare the expression for the momentum of a charge placed in a constant homogeneous magnetic field with that for the momentum of a particle relative to a rotating coordinate system. From (16.29) and (15.33), the former is p = mv + -^ = mvH--^[Hr], (16.36) the latter can bo easily found from (8.5); p = mv' + m [tor]. (15.37) Wo now consider a steadily moving system of identieal charges (for example, an atom or molecule); the nuclei, being heavier, are regarded as fixed. Let us assume that in the absence of any external magnetic field, the motion in the system is known. Then, comparing equations (16.36) and (15.37), it is easy to see that if we consider the motion of these charges relative to axes rotating with angular velocity o> = eH 2mc ’ (15.38) it will not differ from motion relative to fixed axes in the absence of a magnetic field. The equations of motion relative to rotating axes will have their usual form p,=F,, where F, is the force acting on the ith charge in the absence of any external magnetic field, because the correction to the momentum due to the angular velocity to [defined in accordance with (15.38)] will cancel with the correction due to the magnetic field. The magnetic field must be sufficiently weak so that the change in magnetic force with rotation can be neglected. Wo can say, therefore, that, with the application of a constant and nniform weak external magnetic field, a system of charges with identical — ratios begins to rotate with a constant angular velocity eH 1 wl 1 = ~2mc * statement is called Larmor’s theorem, and co^ is the Larmor frequency. Precession of the magnetic moment. If a system possesses a magnetic moment p. for motion which is undisturbed by a magnetic field, then, when a magnetic field is superimposed, this moment will move around the direction of the magnetic field, similar to the free top in eqriations (9.14) (note that to is in the direction of the axis of rotation, i.e., in the direction of the magnetic field). The precession of m.agnetic moment about the field is called the Larmor precession. Magnetic moment in an inhomogeneous field. Let us suppose that the magnetic field possesses a small inhomogeneity. Then, in the equations of motion, the term Sec. 16] THE MAGNETOSTATICS OF POINT CHARGES 143 F = - 13^' = V (Hji) (16.39) denotes the force acting on the moment and tending to move it as a whole. Expanding (15.39) by (11.32), we obtain F == (jjL V) H + [p, rot H]. But for an external field rot H is equal to zero so that the force is F = (pV)H. (15.40) This is the well-known force of attraction to a magnet. It is maximum near the poles of the magnet where the inhomogeneity of the field is greatest. Exercise Study the magnetic moment (i moving in a magnetic field given by the components Hz = — Wo! ~ Hx cos co <; Hy — II ^ sin a t. Consider the cases <0 — o = -» <*> ® 2wc ■ 0 . The equation describing a vector rotating with angidar velocity is, according to Secs. 8 and 9 , d\L dt = [o = ) o), = [Cf. (16.38)], we multiply the equation for gy by ± i and combine with the equation for g* to obtain -gy (g* ± igy) = ± fwolg* ± iv-y) ± V g* — g| — gj • We seek the solution in the form g* ± * gy = .4 ± e *' “•*, and get the following equation for amplitudes A ± : (to — (Oj) A ± = toj "v/ g* - A + A - . Multiplying these equations, we find A + A _ (to- + 0>f A± = g“i "v/ (to—to„)* + tof ■When to = toj (“paramagnetic resonance”), the moment rotates in the plane x y with frequency to^. When to 0 , i.e., in the case of an infinitely slow rotation of the field, the moment strictly follows the field, its direction all the time being the same as that of the field. 144 ELECTRODYNAMICS [Part I] Sec. 16. Electrodynamics of Material Media Field in a medium. We know that material media consist of nude] and electrons, i.e., of very small charges in very rapid motion. There fore, in a small region of a body— a, region having atomic dimensions— all electromagnetic quantities (field, charge density, and current) change very rapidly with time. In two neighbouring small regions these quantities may, at the very same instant, have completely different values. Therefore, if we examine the field in a medium full of charges in detail, then we will observe only a rapidly and irregularly varying function of coordinates and time. Mean values. The inhomogeneities of a field are of atomic dimensions. However, such a detailed picture of the field is not usually of any interest. As usual, in any description of macroscopic bodies, it is essential to know mean values for a large number of atoms. For exam¬ ple, in mechanics, mean density values are used. For this mean tc have any significance, we must isolate a volume of the body contain¬ ing a large number of atoms, determine its mass and divide by the volume. This volume must be so large that the microscopic atomic structure of the substance cannot affect the mean value of the density. At the same time, the mean macroscopic value must be constant over that volume, This will be readily seen from the following. Let the volume be arbi¬ trarily divided into two equal parts. Then the mean for each part should not differ from the mean for the whole volume. Such a volume is termed physically infinitesimal. We shall call it Vq. If we take all its dimensions to be large compared with atomic dimensions, then the mean value should not depend on the shape ot the surface bounding the volume; the latter may be spherical, cubi¬ cal, etc. Besides averaging over volume, it is also necessary to perform averaging over time. The interval of time over which the average is to be taken must be large compared with the times of atomic motions, though still sufficiently small so that the mean values over two semi¬ intervals do not differ from one another. Let the volume have the form of a cube of side a. We shall denote the coordinates of its centre by x, y, z. The time interval, over which the averaging is performed, will be called and the instant corre¬ sponding to the centre of the interval will be denoted by t. The coordi¬ nates of any point inside the cube, relative to the centre, will be called 5, 7], C and instants of time measured from t, will be denoted by 6^. Thus, the limits of variation of the quantities are given by the follow¬ ing inequalities: 2 ^ 2 ’ Sec. 16] EtECTBODYXAMICS OP MATERIAL MEDIA 145 The actual value of any quantity at a definite instant of time is / y+7), 2 + !^, t-{■&). It is related to a mathematically infinitely small volume dV ==d^ dy\ dX, and an interval of time d^. The average value, over a physically infinitesimal volume Fq and interval of time Ig, is obtained if / is integrated over dV dt and the integral divided by Foifl, in accordance with the usual definition of an average: / {x, y, z, t) ^ “A. “/a “/a Va "" J J 2/ + ^> z + ^. < +&). ° -"/a -"/a -“’/a ^Va (16.1 This average value / (x, y, z, t) refers to the point x, y, z, and time t. The electrodynamics of such mean values is termed macroscopic, as opposed to microscopic, which has to do with a field due to separate point charges and a field in free space. The mean thus determined is differentiable with respect to time and coordinates. As parameters it involves the coordinates of the centre of a physically infinitely small volume x, y, z, and the time t. Obviously, we can differentiate with i-espect to these values: / (X, y, z, 1) - ‘U "A ‘■Z. ‘>k ’ vi:: J <*5 1 .<•) J«J» - -“h -V, -“It -w* f(x I, y >r ri, z 1 +^) = (16.2) In other words, the mean value of the derivative of a quantity is equal to the derivative of its mean value. Density of charge and current in a medium. Under the action of the electric and magnetic field, there occurs a redistribution of the charges and currents in any substance. When Maxwell’s equations are averaged, the mean density of the redistributed charge is p and that of the current is j. We shall express p and j in terms of other values which will later make it possible to give the averaged Maxwell equations a very symmetrical form. We define the dipole-moment density in a substance by the follow¬ ing formula: d=JPdF. (16.3) The dipole moment P in unit volume is called the electric polarization of the medium. If the substance is completely neutral, its dipole moment is uniquely determined as ^e,r‘ [see (14.21)]. Going over i to a continuous charge distribution, we write 10 - 0060 146 BiaiCTRODTOAMICS [Part II d=JprdF. Integral (16.3) can be identically written in the form: Jl»dF= - JrdivPdF. (16.4) (16.6) This relationship can most simply be proved by writing in terms of components, for example, J JJ * ( + ^)dxdydz = J J(a!P.) \dydz + + J j a: (Py) I da:dz + j Ja: (P*) dx dy -- j j ^ Pxdxdydz. ri The limits of integration are at the external boundaries of the medium, where the values P*, Py, Pz are zero. This proves (16.5). Comparing (16.4) and (16.6) we obtain J r (div P + p) d F = 0. (16.6) However, since the shape and dimensions of the body are arbitrary, the quantity divP + p = 0 (16.7) must be zero. Thus, the mean density of a charge “induced” by the field is equal to the divergence of the electric polarization vector taken with oppo¬ site sign. In a similar way, wo can express the mean density of induced cur¬ rent. To do this we define the magnetic polarization vector, equal to the magnetic-moment density, as |A = jMdF. (16.8) But the magnetic moment, by definition, is expressed as e,- [r' V'] 2c Applied to the current distribution, this gives p.= / [rljdF 2c We shall now prove an identity analogues to (16.6): fMdF = y/ [rrotM]dF. (16.9) (16.10) Sec. 16] ELECTBODYNAMICS OF MATERIAL MEDIA 147 For this, it is simplest to go over to components: J [rrotM]*dF = J (yrot^M — 2 rotyM)dF = The terms — y~^— integrated by parts. All the integrated quantities become zero when the limits are inserted, so that, in agreement with (16.10), only 2 Mx remains. Now, comparing (16.9) and (16.10) we obtain = J . (16.11) In order to determine j fully, we calculate its divergence and apply the charge conservation law (12.18) written for mean values [see (16.2)]: From (16.11) and (16.12), J is uniquely determined as j=-|^ + crotM. (16.13) Indeed, tliis expression satisfies both equations. Finding the diver¬ gence of both parts of (16.13), we arrive at (16.12), since div rot M==0. 61 ’ Further, substituting the quantity-^— in the left-hand side of (16.10), we get According to (16.3) and (16.4), P is replaced by pr. Ilut, [r, pr] = 0, 6 P SO that the term does not contribute to equation (16.11). The identity (16.13) is thus proved. Averaging Maxwell’s equations. We shall now consider the averaging of Maxwell’s equations. From (16.2), difierentiation and averaging are commutative, so that a bar can simply be put over the first pair in order to denote that they have been averaged: rotE=-|^, (16.16) divH=—0. (16.16) The mean value of an electric field E is called the electric field in a medium. We shall hereafter write it without the bar, which denotes that it has been averaged, taking it for granted that only mean values 10* 148 ELECTROD YNAMICS [Part II will always be taken in a medium. The mean value of the magnetic field is called the magnetic induction and is denoted by the letter B. ft is all the more unnecessary to write a bar over it because the concept of induction, which is not equal to field, makes sense only for a medium. The asymmetry in the terminology for electric and magnetic fields will be explained later. In this notation, the first pair of Maxwell’s equations takes the following form: rotE= - (16.17) divB:.-0. (16.18) Now let us average the second pair of Maxwell’s equations: , Y7 1 SE , in— divE = 47rp. (16.20) We substitute p and j from (16.7) and (16.13) and rearrange the terms somewhat, obtaining the two following equations (the bars are again omitted): rot(B -47tM) + 47tP), (16.21) C O t div (E + 47rP) = 0 . (16.22) We introduce the following now designations: E-f-47tP-=D. (16.23) D is called the electric induction. Further, B-—47iM=^H. (16.24) II is called the magnetic field in a medium, which, therefore, does not equal the mean value of the magnetic field in a vacuum. In the notations (16.23) and (16.24), the second pair is written similar to the first pair: rot II (16.25) divD-0. (16.26) The similarity between (16.26) and (16.18) explains why it was convenient to call the mean value of the magnetic field the magnetic induction: here, both electric and magnetic induction vectors have no sources in the medium. The similarity between (16.25) and (16.17) justifies the term magnetic field given to the vector H (16.24). The incompleteness of the system of equations in a medium. Thus, due to a suitable system of notation, the first and second pairs of Sec. 16] ELECTRODYNAMICS OF MATERIAL MEDIA 149 Maxwell’s equations in a medium have, as it were, become more symmet¬ rical than those in a vacuum. But we must not forget that this system has now ceased to be complete: as before, there are eight equations (of which only six are independent) and twelve unknowns B, E, D, H (with three components for each vector). Consequently, the system (16.17), (16.18), (16.26), and (16.26) cannot be solved until a relationship is found between inductions and fields. This relationship cannot be obtained without knowing the specific structure of the material medium. Dielectrics and conductors. We shall consider, first of all, how charges behave in a medium in the presence of a constant electric field. The field will displace the positive charges in one direction, and the nega¬ tive ones in another. As a result, polarization P will arise. Two essen¬ tially different cases can occur here. 1) Under the action of the field inside the body, a certain finite polarization P, dependent on the field, is established. This polariza¬ tion may be represented vividly (though in very simplified fashion!) as a displacement of charges from the equilibrium positions which they occupied in the absence of the field to new equilibrium positions— much like the way a load suspended on a spring is displaced in a gravi¬ tational field. If a finite polarization (dependent on the field inside the body) is established, that body is called a nonconductor or dielectric. 2) In a constant electric field acting inside the body, the charges do not arrive at equilibrium, and a definite rate of polarization increase, dV , is established. In this case, through every section of the conductor Ot gp perpendicular to the vector , there pass electric charges or, what amounts to the same thing, a flow of current. From equation (16.13), gp the derivative may indeed be interpreted as a current component. As regards the second current component, c rot M, it relates to the instantaneous value of quantities and cannot characterize the change of anything with time. For this reason, the classification of bodies into conductors and dielectrics is obtained from the behaviour of the SP quantity . The displacement of charges under the action of a field can roughly be likened to a load falling in a viscous medium with friction, when, as is known, a definite speed of fall is established. A medium, in which a constant electric field produces a constant electric current, is called a conductor. If a constant electric field is produced in free space and a conducting body of finite dimensions is introduced into it (for example, a conduct¬ ing sphere or ellipsoid), the charges in the body will be displaced so that a field equal to zero will be established inside the conductor. For this, the mean charge density inside the body must also equal zero. 160 ELECTBOD YNAMICS [Part n because the lines of force of the field originate and terminate at charges. Under the action of such a field, the charges inside the conductor would be displaced. This means that an equilibrium will be established inside the conductor only when all the induced charges emerge to the surface. They will be distributed on the surface of the body so that the mean field inside the conductor is zero, and the lines of force outside the conductor will arrive normal to every point of the surface. Continuous current in a conductor. A continuous ciurent can flow in a conductor only along a closed conducting circuit. And the electric field must always have a component along the direction of the circuit. Then charges of given sign wiU always move in one direction, thereby producing a closed current. The work performed by unit charge in moving around the circuit is called the e.m.f. acting in the circuit: e.m.f.= jpidl. (16.27) This formula differs from (12.1) in that E denotes the field acting inside the conductor. External sources ol e.m.t. In a conductor, a constant e.m.f. can only exist at the expense of some external source of energy, for example, a primary cell. When a current passes in the circuit of the cell, ions are neutralized on the electrodes, thus 3 delding the source of energy that maintains the e.m.f. If, as usual, we put E == — yq), then the expression for e.m.f. will be e.m.f. = -JV 9 dl =-dij + ^dz) = —Jd9 = 9i —92. (16 28) Therefore, the e.m.f. may be defined as the change in potWtial in going round a closed path. Thus, the potential is not a unique function of a point: for each traverse, it changes by the value of the e.m.f. in the circuit. The magnetic properties of bodies. We shall now consider the magnetic properties of bodies. In a constant magnetic field, a definite equilibrium state will always be established in the medium. Here we must distinguish between the following two cases. 1) In the absence of a field, the atoms or molecules of a substance possess certain characteristic magnetic moments that differ from zero. As was shown in the previous section, the energy of every separate elementary magnet in a magnetic field is — [a H. Hence, the energy of elementary magnets that have a positive moment projection on the field is less than the energy of elementary magnets with a Sec. 16] ELECTRODYNAMICS OE MATERIAL MEDIA 161 negative moment projeetion. Atoms and molecules are in random thermal motion. As a result of this motion and of the action of a magnetic field, an advantageous energy state is established in which positive moment projections on the field predominate. For more detail about this equilibrium see Part IV. It will be noted that the projection of an isolated magnetic moment on the magnetic field is constant—^it merely performs a Larmor precession around the field. But the interaction between molecules disturbs the motion of separate moments, and results in the establish¬ ment of a state with a mean magnetic polarization other than zero. 2) The atoms or molecules of a body do not possess their own magnetic moments in the absence of a field. As was shown in the previous section, when an external magnetic field is applied, the motion of charges in atoms or molecules changes due to the Larmor precession. Indeed, a precession of angular veloc- eH ity (0 = is superimposed on motion undisturbed by a magnetic field. In exercise 4 of this section it will be shown that this precession leads to the appearance of a magnetic moment in a system of charges. We shall only note here that the direction of the magnetic moment induced by the field must be in opposition to the direction of the magnetic field; this follows from Lenz’s induction law. Indeed, an induced current produces a magnetic field in a direction opposite to that of the inducing field. A substance in which an external magnetic field produces a resultant moment in the same direction, is called paramagnetic. If the magnet¬ ization is in the opposite direction to the field, the substance is dia¬ magnetic. Ferromagnetism. There are crystalline bodies in which the magnetic moments are aligned spontaneously, i.e., in the absence of any external magnetic field. Such bodies are called ferromagnetic. The magnetic polarization of the body itself is related to the directions of the crystalline axes. For example, in iron, whose crystals have cubic symmetry, the intrinsic magnetization coincides with one of the sides of the cube. This direction is called the direction of ready magnet¬ ization. In order to deflect the magnetic polarization from the direction of ready magnetization, work must be performed. A single crystal of a ferromagnetic substance will be magnetized so that the resultant energy is a minimum—equilibrium always corresponds to minimum energy. However, it does not necessarily follow from this that all of the single crystal is magnetized in one direction; in this case it will possess an external magnetic field whose energy is ^ H^dV. This quantity is always positive and increases the total energy. But if the single crystal is divided into regions or layers whose magnetization alternates in direction, then the 162 BltECTBOD YNAMICS [Part II external field can be eliminated since neighbouring layers (or, as they are called, domains) produce fields of opposite sign. In the tran¬ sition region between domains, the polarization gradually turns from the direction of ready magnetization in one domain to the reverse direction in the other domain. Clearly, if a certain direction is the direction of ready magnetization, then the directly opposite direction also possesses this property. The structure of the transition region has been studied theoretically by L. D. Landau and E. M. Lifshits. The domain structure of crystals was later demonstrated experi¬ mentally. If a very thin emulsion of particles of a ferromagnetic substance is spread over the smooth surface of a ferromagnetic single crystal, the particles will be distributed along lines where the inter¬ faces between domains intersect the surface of the crystal. Since, between domains, the polarization is deflected from the direction of ready magnetization, it is necessary to perform work to establish the transition region. Summarizing, if the whole single crystal consists of one domain, its energy increases at the expense of work done in creating an external field, equal to J Il^d V ; if, however, the crystal consists of many domains, the energy increases at the expense of the additional energy of the transition regions. 'I'he equilibrium state will be that state in which the energy is least. The energy of a field increases with the volume it occupies, that is, as the cube of the linear dimensions of the crystal. The energy of the transition regions increases in proportion to their total area. In a crystal of sufficiently small dimensions, there can exist only one transition region whose area is proportional to the square of the linear dimensions of the single crystal. Therefore, in such a small crystal, the volume energy changes according to a cubic law with respect to dimensions, while the surface energy varies according to a square law. In a sufficiently small crystal, the volume energy becomes less than the surface energy; such a crystal is not separated into domains but is magnetized as a whole. This has been experi¬ mentally established in crystals with dimensions of 10“*-10'® cm. The thickness of domains in large crystals of appropriate ferromagnet¬ ics is of the same order. An example of the shape of a domain as proposed by Landau and Lifshits is shown in Fig. 22. The arrows denote the direction of the polarization in each I ill 1 1 domain. The serrations at the boundary almost completely destroy the external magnetic field; Fig. 22 the lines of magnetic induction inside the crystal are closed tlmough them, and do not emerge. Fig. 22 The magnetization of a ferromagnetic in an external field. If a magnetic field is applied to a ferromagnetic crystal in the direction of ready magnetization, then those regions, for which the polarization Sec. 16] EUECTBODYNAMICS OF MATEBIAL MEDIA 163 is in opposition to the field, are contracted with displacenaent of the interfaces and may disappear completely in a comparatively small field. Then the crystal is magnetized to saturation. In order to magnetize the crystal to saturation in a direction that is not coin¬ cident with the initial direction of polarization in the domains, con¬ siderably larger fields are required. In a polycrystaUine body, such as ordinary steel, the separate single crystals are oriented more or less at random relative to one another. In any case, the directions of ready magnetization are not the same for the separate crystals. When an external magnetic field is applied, the different crystals are magnetized differently and the magnetization curve is not as steep as is possible in the case of a separate crystal. The magnetic interaction between separate crystals results in a definite magnetic polarization remaining after steel has been magnetized and the field subsequently removed. This is what is known as hysteresis. Magnetic interaction of atoms. Let it be noted, in addition, that the magnetic interaction between separate elementary (atomic) magnets is not at all adequate in explaining the cause of ferromagnetism. The energy of interaction between two elementary magnetic moments is of the order IQ-i® erg, while the energy of thermal motion at room temperature is about 10 erg (see Part IV). This is why random thermal motion should destroy the orderly magnetization already at a temperature of about 1° above absolute zero. Actually, ferro¬ magnetism of steel disappears in the neighbourhood of 1,000° above absolute zero, thus corresponding to an interaction energy between elementary magnets in the order of 10~^® erg. Ferromagnetism is of quantum origin and cannot be explained with the aid of classicial analogues. The relationship between fields and inductions. A substance is always in equilibrium in a constant external magnetic field. To this equilibrium there corresponds a very definite induction and polari¬ zation. In a weak field, the relationship between the quantities is linear. For this reason, the magnetic induction is expressed linearly in terms of the magnetic field in a medium: B = xH. (16.29a) In a dielectric, where a static equilibrium polarization corresponds to a definite electric field, there is a similar relationship for weak fields. D = sE. (16.29b) The quantity x is called the permeability and s is the dielectric constant. It should be noted that in ferromagnetics the region for which a linear law is applicable has an upper limit of not very large fields 154 ELECTBOD YK AMICS [Part II (10®-10* CGSE), since saturation sets in; in diamagnetic and paramagnetic substances at room temperatures a linear law applies for all actually attainable fields. The vector nature of electric and magnetic fields. The question may arise: Why is magnetic induction expressed linearly solely in terms of magnetic field, while electric induction is expressed solely in terms of electric field? In order to answer this question we must examine the vector properties of electromagnetic quantities in more detail. Two separate systems of rectangular coordinates exist in space; a right-hand system and a left-hand system. They are related to each other like left and right hands, if the thumbs are in the direction of the x-axes, the forefingers along the y-axes, and the middle fingers along the z-axes. It is obvious that no rotation in space can make these two systems coincide. However, one system transforms to the other if the signs of the coordinates in it are reversed. Of course, both coordinate systems are completely equivalent physi¬ cally. The choice of any one of them is completely arbitrary. Therefore, the form of any equation expressing a law in electrodynamics should not change under a transformation from a right-hand to a left-hand system. Let us now take Maxwell’s equation (12.24). In order to perform a transformation to another coordinate system, it is sufficient to change the signs of the coordinates. This changes the sign of the vector operation rot, because this operation denotes a differentiation with respect to coordinates. What happens, then, to the electric field components ? Since only one of two vectors is differentiated with respect to the coordinates, namely E, the sign of one of them must change in order to retain the form of the equation. It is easy to see that vector E will change sign. Indeed, the right-hand side of equation div E = 4 u p is a scalar and does not change in sign. On the left- hand side, the sign of the div operation changes and, hence, the signs of all the components of E must also change. Therefore, the com¬ ponents of the magnetic field do not change sign in a transformation from a left-hand to a right-hand system. In a rotation of a coordinate system, the projections of any vector are transformed by the same equations of analytical geometry as the coordinates. As was shown, the change of sign for all three coordi¬ nates is not equivalent to any rotation. It turns out that some vectors, such as E, behave quite similarly to a radius vector r; when the signs of the components of the radius vector r are changed, the signs of all the components of E also change. Other vectors, such as H, behave like a radius vector under coordinate rotations, and not like a radius vector in the transformation from a right-hand to a left-hand system. Vectors that behave like E are called true or polar vectors, while those behaving like H are called pseudovectors or axial vectors. Sec. 16] ELECTRODYNAMICS OF MATERIAL MEDIA 165 Velocity, force, acceleration, current density, and vector potential are, in addition to the electric field, real vectors while magnetic moment, angular momentum and angular velocity are pseudo¬ vectors. The fact that angular momentum is a pscudovector can easily be seen from its definition: M = [rp]. Both factors of the vector prod¬ uct, r and p, change signs, so that M does not change in sign. A pseudovector cannot be linearly related to a real vector in electro¬ dynamics because the sign in any such equality would depend on the choice of coordinate system, which contradicts physical facts. For this reason the vectors B and H, D and E appear separately in the linear laws (16.29a) and (16.29b). The equations for conductors in a constant field. We shall now consider the equations of electrodynamics of constant fields for conductors. As has already been indicated, it is not a constant value of polarization that is established in a conductor in a constant field. but a constant rate of increase of polarization . This quantity has Ot ^ J) the meaning of current density j'. The derivative appearing on the right-hand side of Maxwell’s equations, may be replaced thus: 3D . 6P = 47tJ' , (16.30) because the field is constant. Here, the current j' is also continuous. The magnetic field is a pseudovector and camiot be linearly related to the current density. We note that for metals the linear relationship between field and current (Ohm’s law) does not break down, no matter how strong the field. The quantity is called the specific conductance, or conductivity. In the CGSE system its dimensions are inverse seconds. For metals ct is of the order 10*^ sec“^. Slowly varying fields. So far, an electromagnetic field in a medium has been regarded as strictly constant with time. But if the field varies sufficiently slowly with time, it may also be considered as constant. Let us give a general criterion whereby we can say what field may be regarded as slowly varying. We assume that a constant field is switched on at some initial instant of time < = 0. A stationary state is not established in the medium at once but only after a certain interval of time 0 has elap.sed. If, for example, the medium is a dielectric, then, in that time, a def¬ inite polarization is established corresponding to the given field; in a metal, 0 characterizes the time taken for a constant current to be established. 0 is called the relaxation time. If, during the relax¬ ation time, the field changes by only a small fraction of its value it can be regarded as constant within the accuracy of that small fraction. In other words, the criterion of slowness of variation of a 156 ELECTKOD YNAMICS [Part II field is this: within the relaxation time a stationary state, correspond¬ ing to the given value of field, has time to establish itself in the medium. Such fields are termed slov/ly varying. For them, the same values of permeability, dielectric constant, and conductivity can be sub¬ stituted into Maxwell’s equations, as for constant fields. Lot us write down Maxwell’s equations for a slowly varying field in a conductor. In the exi)ression ai) bt dTj bt A 47t - - bt _SE bt 47tcrE (16.31) the first term can be neglected in the majority of cases because it in no way exceeds y-; if, as occurs in metals, a is of the order of 10^^, then aE "p- EjQ. Whence Maxwell’s equations are obtained for a slowly varying field in a conductor: X Ti 47rj' 47rcrE rot H ^ - =-, C C (16.32) rotE - , c bt (16.33) div X H = 0. (16.34) This system is complete and, together with the boundary con¬ ditions (see exercises 1 and 5 in this section), is sufficient for the deter¬ mination of slowly varying fields in a conductor. Rapidly varying fields. \jet us now consider the case of rapidly varying fields, i.e., fields which change more rapidly than the relax¬ ation process or the establishment of a definite stationary state in the medium. Then the state of the medium depends not only on the instantaneous value of the field, but also on its values at previous instants of time; in other words, it depends on the way in which the field changes with time. Such a relationship is very complicated in the general ease. It is simplified if the field is weak ; then, at any rate, wo may e.xpect the relationship to be linear. Expansion in harmonic components. Let us examine the general form of the linear relationship. To do so we represent the field as follows: E(<)=^EfcCos(plicablo to the case of rapidly varying fields, if the relationship between field and time is consi 0, () t V t then, for a periodic variation of B, the heat generated in one period is equal C H d to J —^- , where the integral is taken over one period. 18) Show that if e ( then t)n the right-hand side there is the expression- - — to (ej EE* X 2 HH*) . From this it can bo sotm that ej 0 and Xs > 9, since the energy of the field is absorbed by the medium. 19) Calculate tho dielectric constant of a medumi, considering that all the chargt's in it arc conncctctl by elastic forces with tho equilibrium ixjsitions. The characteristic oscillation frctiuoncy of tho charges is o>g and tho frequency of tho field is w. Tho raditis vector of a charge satisfies the differential equation m ( r -1- r) « e E* e— iwt = e E. Its solution has the frequency of tho external field and may bo written as eE I’ho polarization can bo obtained from this by multiplying by tho munber of charge's in unit volume N and by e. Since the induction D is elane x = ct, because E (.c — ct) =E(0) on that plane. We can also say that the plane on which the field E is equal to E (0) is translated in space through a distance ct in a time t, i.e., it moves with a velocity c. The same applies to any plane x = Xq, for which there was some value of field E (Xq) at the initial instant of time. To summarize, all planes with the given value of field are propagated in space with velocity c. Therefore, the solution E (a: — ct) is called a travelling plane wave. We note that the form of the wave does not change as it moves; the distance between planes x — x^ and x = Xn, for which E is equal to E (Xi) and E (x^), is constant. This result holds for any arbitrary form of wave, i^ro- vided it is travelling in free space. llepcating, the velocity of propagation of a wave in empty sjiace does not depend on its shape or amplitude and it is equal to a universal constant c. The transverse nature of waves. The elec¬ tric and magnetic fields, as we have seen from (17.19), are perpendicular to the direction of wave propagation, as well as to each other. This is why it is said that elec¬ tromagnetic waves are transverse (as op- Fig. 24 posed to longitudinal sound waves in air, for which the oscillations occur in the direction of propagation). The direction of propagation, the electric field, and the magnetic field are shown in Fig. 24. In it, n is a unit vector along the x-axis. In future it will be sufficient to take only one component of the electric field. For this it is necessary to take one of the coordinate axes, for example the y-axis, in the direction of the electric field, which in no way limits the generality. This is shown in Fig. 24. 166 ELECTRODYNAMICS [Part II The a:-coordinate will be written in the form x = rn, so that Ey = Ay{tn — ct). (17.20) But in this notation it is not necessary to relate the vector n, in the direction of propagation to the a;-axis. A solution with argument of the form (17.20) is applicable to any direction of n, provided, naturally, that n, E, and H are mutually perpendicular. The momentum density of the wave [see (13.27)] is equal to 1 47CC [EH] 1 4xc A} and is directed along n. The energy density is KA + W 1 Sir ~ 4n It differs from the momentum density by the factor c. Tliis, as we shall see later, is very essential for the quantum theory of light. Pressure ol light. If a wave falls on an absorbing obstacle, for example, on a black wall, and is not reflected, then its momentum is transmitted to the wall in accordance with the conservation law. But momentum transmitted to a body in unit time is, by Newton’s Second Law, nothing other than force. It follows that there is a force of - 4 -^- for every square centimetre of the absorbing barrier, upon which the wave is normally incident. Force referred to unit surface is, by definition, the pressutr of the electromagnetic wave on the barrier. Consequently, electrodynamics predicts the existence of light pressure. This was observed and measured by P. N. Lebedev. Harmonic waves. A special interest is attached to travelling waves for which the function E (x — d) is harmonic. The most general harmonic solution is of the following form: E =Rc|re—'“(‘--r)j , (17.21) where the symbol Re {} denotes the real part of the expression inside the braces, F is a conqilex vector of the form -{-i [Cf. (7.14c)], and to is the wave frequency in the same sense as in equation (7.3). to is the number of radians per second by which the argument of the exponential function changes. The wave vector. The vector to — is called the wave vector. It is C denoted by the letter k: k s to — . C (17.22) Sec. 17] PLANE BLBCTBOMAGNETIO WAVES 167 The geometric meaning of k is easy to explain. We define the wave¬ length, i.e., the distance ^ r-n in space at which E assumes the same value. Let the required wavelength be X. Then . X Ar*n 10) — 10)- 2 m e ‘ = e ' = e (17.23) because the period of the function e‘* is equal to 2 n. Hence, X = (17.24) Comparing the wavelength with tlie wave vector, we obtain k = 11, X = . (17.26) Polarization of a plane harmonic wave. Let us now study the nature of the oscillations of an electric field. To do this, we write the vector F in the form F = Fi -f iF„ (El — t R,) 6'“. (17.26) We choose the phase a so that the vectors Ej and Eg are mutually lierpendicular. We multiply equation (17.26) by e-** and square. Then we obtain (El —iEg)^ = 7^2 (17.27) We have taken advantage of the fact that Ei and Eg are perpendic¬ ular. Because of this (Ej — iEg)* is a purely real quantity. Therefore, the imaginary ])art of the right-hand side of expression (17.27) must be put equal to zero. Representing as cos 2 a — i sin 2 a, we obtain or — {FI — FI) sin 2a -f 2 (Fi Fg) cos 2a -- 0 , 2 (F, Fj) F(-Fl ’ (17.28) whence the angle a is determined for the given solution (17.21). It is now easy to express Ei and Eg. Indeed, from (17.26), Ei — — Eg = (Fi + i Eg) e-'“ = Fi cos a -}- Fg sin a — i (Fi sin a - Fg cos a), so tliat El = Fi cos a -f Fg sin a, 1 Eg = Fi sin a — Fg cos a. J (17.29) We now include a constant phase in the exponent (17.21) and, for short, put 163 ELECTRODYNAMICS [Part TI Then, in the most general case, the electric field for a plane harmonic wave will bo E = Re {(El — i Ej) = Ejcos t{; + E 2 sin •]). (17.31) Here, the vectors E^ and Eg are defined as perpendicular. Let us assume that a wave is propagated along the x-axis. The t/-axis is directed along Ej, and the 2 -axis along Ej. Hence, from (17.31), we obtain Ey = El cos , Ez = E^ sin i};. (17.32) Let us eliminate the phase (I*- We divide the first equation l>y E^ , the second by E ^, square and add. Then the phase is eliminated and an cipiation relating the field components remains: + E} = 1 . (17.33) It follows that the electric lielfl vector describes an ellipse in the // 2 -i)lano moving along the x-axis with velocity c, and passes round the whole ellipse on one wavelength. Relative to a fixed coordinate system, the electric field vector describes a helix wound on an elliptic cylinder. The pitch of the helix is equal to the wavelength. Such an electromagnetic wave is termed elliptically jiolarized. It represents the most general form of a plane harmonic wave (17.21). If one of the comi)onents i.s equal to zero, for example Ei = 0 or E 2 = 0, then the oscillations of E occur in one plane. Such a wave is termed plane polarized. When Ey is equal to E ^, the vector E describes a circle in the ;iy 2 -plane. Ile])ending on the sign of Ez , the rotation around the circle occurs in a clockwise or anti¬ clockwise direction. Accordingly, the wave is termed right-handed or left-handed polarized. These waA'cs are shown in Fig. 25. For the same A-alue of phase d/, the rotation occurs either in a clockwise or anticlockw'ise direction. The sum of two waves of equal amplitude, which are circularly polarized, gives a plane polarized wave. The relationship between their phases determines the plane of polarization. Thus, if the waves show'll in Fig. 25 are added, the osciUations Ej and —Eg mutually cancel and only the plane polarized oscillation Ej remains. In turn, a circularly polarized oscillation is resolved into two mutually perjiendicular plane oscillations. Certain crystals, for example tourmaline, are capable of polarizing light. Sec. 17] PLANE ELECTKOM.AGNETIC WAVES 109 IJnpolarized light. In nature, it is most common to observe un¬ polarized (natural) light. Naturally, such light cannot be strictly monochromatic (i.e., possessing strictly one frequency to), for, as we have just shown, monochromatic light is always polarized in some way. But if w'e imagine that the components Ej and Eg in Fig. 25 are not related by a strict phase relationship (17.32), but randomly change their relative phases, then the resultant vector will also change its direction in a random manner. However, for this, it is necessary that the oscillation frequencies should vary in time witluTi some interval A to, since the difference of phase between two oscil¬ lations of strictly constant and identical frequency is constant. The propagation ol light in a medium. We shall now consider the question of the propagation of light in a material medium. At the end of the preceding section we said that the quantities s and x have meaning only for oscillations of a definite frequency (o. To simplify notation, we shall not use the symbol for a real part Re {}, remembering that the real part is alw.ays taken. Since all the quanti¬ ties depend on time according to an c“'“' law, the derivative reduces to a multiplication by — i w. Then the .system of Maxwell’s equations can be written in the following form: rotH= --^eE, (17.34) div E = 0, (17.35) rotE = -^xH, (17.3«) div H = 0 . (17.37) Once again we look for a solution in the form of a plane wave. Since the time relationship is already eliminated, all the quantities depend only on one coordinate, for example, upon x. From (17.35) and (17.37), it follows that SEx dHx f, Bx ’ Bx or Ex = 0, Hx = 0, because a solution that is constant over all space does not repre.sent any w’ave. Thus, the waves are transverse. Equations (17.34) to (17.37) are satisfied if we substitute Ey = E {x), E^ = 0, Hy = (), Hz = H (x), or, in other words, if the electric field is directed along the y-axis and the magnetic field along the z-axis (a right-handed system). 170 E LEOTROD YNAMICS [Part II Indeed, there then remain the following equations: dll dx dFj dx ivi c i . c ^ c All three phases must coincide over the whole interface, whence 'Ja sin 0 = vb sin ft (the law of refraction), 0 = 0i (the law of reflection). 172 ELECTKODYXAMICS [Part II Taking into account that II = yll [this is easily obtained from (17.39) and (17.41)], wo write the boundary conditions (see e.xercise 1, Sec. Hi); v’ (I'J sin 0 — El sin 6) = E^ sin & , E cos 0 d- E-i cos 0 = E^ cos ft , Va(&’ — El) = vfjA’j , where E, A’, and E„ aro tlio electric fields in the incident, reflected and refract¬ ed waves. Wo can see that, by eliminating A?, K, and E^ from tho.se conditions, wo again obtain (he law of refraction. The ratio of the amplitudes is El _ tan (0 — ft) E tan (0 -b ft) (I) If 0 -1- ft = "iy, then El = 0 and reflection does not occur. (How can this he vorifiotl by double reflection ?) In case. If we must write down the boundary conditions and obtain (he eipiation El E sin (0 — ft) sin (0 -f- ft) (II) (I) and (I'l) are called (ho Frc'snol equations. 2) In the case — sin 0 > 1 (total internal reflection), show that instead V|, of equation (1) and (If), a reflection coefflcient of unity is obtained. B’ind at wliat ilepth (ho wave, passing in the medium 6, is attenuated c times. 3) B'ind the frequency of electromagnetic oscillations in an infinite square prism with perfectly roiloct.ing walls, assuming a longitudinal electric field constant along the k'ngth of the prism. Consider that the field inside the prism doe.s not become zero. Wo must consider that the tangential component of the electric field at th(' walls of (ho prism is e(|ual to zero, so that the normal component of the I’oynting wet or U should become zero. 'Pho solution to Maxwell’s ccpiations can he obtained from the potential Ax = ..4„ sinsin-^^e— ,Ay = -1- = a a --•'fi = 0 (the .i-coordinate is taken along the axis of the prism). It is of the form Ex - A'o S'*' sin e , a a IIy = I/p cos sin —- 6“, ‘ a a IIX ---- — Hfl sin —~ j.Qg JUL e a a on the condition that 27t2 E„ = lip and c* (a is the side of the square). 4) Solve the same problem for a travelling wave in the prism (a waveguide). The field in a w’aveguide is not zero anywhere except at the walla. The form of the vector potential in the previous problem suggests one of the following possible solutions: Sec. 18] TRANSMISSION OF SIGNALS. ALMOST PLANE W.AVES 173 Ax = A^e ' sin Ct Or = *4oye~' cos -^sin-^, -4z =

- We now introduce a nexv integration variable co — coq. Then the integration can be easily performed and the field reduces to the follow¬ ing form: Ao >/2 = FqC -i(Mo/-CoJr). . (18.4) Sec. 18] TRANSMISSION OP SIGNALS. ALMOST PLANE WAVES 175 The shape of the signal. Let us now examine the expression obtained. It consists of two factors. The first of them, repre¬ sents a travelling wave homogeneous in space with a mean “carrier" frequency coq- However, the amplitude of the resultant wave is no longer constant in space because of the second factor: where the designation g and ij* are obvious from the equations. This factor has a greatest maximum at t, i.e., when the argument of the sine and the denominator are equal will be the less the greater their number (Fig. 27). The greatest maximum is equal to A CO (since ■ =1 for = 0). This maximum is not situated at a fixed place, but moves in space with a velocity d<^ (18.5) because, from the definition of the point of maximum t}/ = 0, it follows that dk t — vt. to zero. The other extremes As we indicated at the beginning of this section, a signal can be transmitted from one point of space to another by means of a displace¬ ment of the maximum, since this maximum is distinguished from other maxima. A disturbance of this kind concentrated in space is called a wave packet. The propagation of a signal of arbitrary shape. A wave packet need not necessarily have the form shown in Fig. 27. By choosing a rela¬ tionship for Eq (w) other than that in equation (18.1) (i.e., by choosing not a constant amplitude in the interval A w, but a more complicated frequency function), the shape of g (({/) can be changed. For instance, the resultant amplitude may have the shape of a rectangle, so that the transmitted signal will resemble the dash in the Morse code. If the frequency coq is within the radio-frequency range, then the signals can follow upon one another within audio frequency, in this way repro¬ ducing music or speech. The frequency range and the duration of the signal. In order to trans¬ mit a signal, it is always necessary to choose a range of frequencies. 176 ELKCTBODYNAMICS [Part IE Let US determine this range. Suppose that the receiving device is situated at some point a: = const. The width of the received signal can be seen from Fig. 27. In units of 4<, it is equal to 7t in order of magni¬ tude. Therefore, the duration of the signal is determined from the equation AtJ; = ■ A<~Tr. In other words, the duration of the signal A< is related to the frequency interval Aw necessary for its transmission by the expression Aw-Ai^ —2tc. (18.6) It should be noted that this estimate refers only to the order of magnitude of A w and At. The determination of AiJ^ is, to some extent, arbitrary. In certain cases Aw-A# > 2 tt, so that the estimate (18.6) is a lower figure. If a radio station is required to transmit sounds audible to the human ear, then the quantity A< is not greater than 0.5 x 10~^ sec, since the limit of audibility is 2 x 10^ oscillations per sec. From this Aw = 2 7t- 2.10L The range of Aw is always less than the “carrier” frequency Wq which, even for the longest-wave transmitting stations, is not less than 1.5 X 10® X 2 tc. In practice, an interval of A w three or four times less than the value given is quite sufficient, since clipping off the very highest frequencies in music, singing or speech does not introduce any essential distortion. Television transmissions require a considerably greater frequency interval, because an image must be reproduced 25 times every second; and, in turn, the image consists of tens of thousands of separate signals (points). As a result, the carrier frequency is about 2 7ux6x 10'^, corresponding to the metric band of radio waves. Such waves are prop¬ agated over a relatively small radius. They are screened by the cur¬ vature of the earth’s surface like light. The relation (18.6) is always correct in order of magnitude; therefore, for distant television trans¬ missions, it is necessary to have either relay stations, very high-placed transmitters, or cable lines. Phase and group velocity. We shall now consider in more detail the velocity with which signals are transmitted. From (18.5), the velocity of a wave packet is d(^ It differs from the propagation velocity of the constant phase sur¬ face, which is expressed in terms of frequency and wave number as Sec. 18] TBANSMISSIOUf OF SIGNALS. ALMOST PLANE WAVES 177 Indeed, the expression for a travelling monochromatic wave can be written in the following form; E . Comparing this formula with the general expression for a traveUmg wave E-—E (x — id), we arrive at (18.7). The velocity of the wave is u, and not c, because (18.7) is by no means necessarily related to the pro¬ pagation of a plane wave in a vacuum. = -^ is called the 'pJmse velocity of the wave; v is called the group velocity of the wave packet obtained by superimposing a group of waves. In a vacuum, v and u coincide because u^ — ck. However, if there is dispersion, i.e., a dependence of the refractive index on the frequency, then w = -^ A: so that v^k. The group velocity may be regarded as the velocity of propagation of a signal only when it is less than the velocity of light in free space c. If the expression (18.5) formally gives t; > c, we cannot avoid a more careful analysis that takes into account absorption. As a result, it turns out that an electromagnetic signal in the form of a very weak precursor is propagated with a velocity c, but the major portion of the wave energy arrives at the point of reception with a lesser velocity (see A. Sommerfeld, Optik, Wiesbaden, 1950). As an example of the calculation of group velocity we shall take the dependence of frequency on the wave vector in the form: co2 = 4- k^. This form is obtained for a waveguide (see exercise 4, Sec. 17, or exercise 19, Sec. 16, in the limiting case of extremely large frequencies). Whence the group velocity is _ c^k Ct> and since ck < to, we have v c. We note that uv—c^. In vector form the group velocity is defined as follows: 9 dh A Ao; = -- Ak-Ax~2Ti; . (18.9) This means that if we want to limit the extent of an electromagnetic disturbance to a region Ax, we must perform a superposition of mono- 2 TC chromatic waves in the interval of values k of order -r—. In three Ax dimensions (18.9) is rewritten thus: Akx • Axr^ 2 t: , Aky ■ Ay 2ti , (18.10) Akz- Az —^27r. The limiting accuracy of radiolocation. We shall explain the relations (18.10) by means of a graphic example. Let us suppose that an electro¬ magnetic wave has, in some way, to be bounded on the sides, as in the case of a radiolocation (radar) beam. Let us find the greatest accuracy with which the locator can register the position of an object at a dis¬ tance 1. Obviously, this accuracy is given by the transverse diameter of the beam d at a distance I from the locator. Let the frequency at which the locator works be equal to w, then the corresponding wavelength is X = • If the electromagnetic wave were to be propagated in unbounded space it would have an accu¬ rately defined wave vector k = -^n (18.11) (n is a unit vector in the direction of the beam). If the wave has a cross section d, then k can no longer be regarded as an accimately defined vector along n. In order to write down an expression for the electromagnetic wave at any point in space occupied by the beam, it is necessary to take a group of plane waves whose vectors k lie inside a cone described by a certain angle of flare. The maximum deviation of the wave vectors of these plane waves from the mean vector k, determined from (18.11), will bo called kj^. Here, we do not have in mind a cone with a sharply bounded surface, but only an estimate of the angular flare ofthe beam. According to (18.10) is related to the whole cross section of the beam by the following relation: d • 2tz . ( 18 . 12 ) Sec. 18] TRANSMISSION OF SIONATS. ALMOST PLANE WAVES 179 Here, we have put Ax=d, ^K=2 because the inaccuracy obtains on both sides of the axis of the beam. The dimensions of the reflector of the locator itself can be ignored if the diameter of the beam is considered at a great distance; and this is of practical interest. In other words, d is determined only by the relationship (18.12) and is independent of the dimensions of the re¬ flector. The divergence of the beam of rays at every point is measured by the ratio . For this reason, the ratio of the cross section of the beam d to the distance from the locator I cannot be less than the quantity 2fci k • d ^ 2 * 1 . I ^ k • (18.13) This relationship is shown in Fig. 28 for the limiting case of the equality. However, it must be borne in mind that it is not in reality an equality but an estimate of order of magnitude. (18.12) is also approx¬ imate and the symbol > must be written in it. Thus, we have obtained two estimates for kx'- kx> ^ (lower estimate) and, from (18.13), ^^ ~ (upper estimate). Eliminating kx from these estimates, we obtain or finally JL lx ~ d d>VT\ (18.14) For example, if Z=100 km and X=1 m, then the position of the object caimot be determined with an accuracy exceeding 320 m. This is why the dimensions of the reflector could be neglected in the estimate. The limit of applicability of the concept of a ray. Equations (18.10) indicate within what limits the concept of a ray is applicable in optics. Obviously, one can talk about a ray in a definite direction only when Ak<^k, (18.16) i.e., when the transverse broadening of the wave vector is considerably 2 TC 2 7C less than the wave vector itself. But and k'^ —— so that (18.16) is equivalent to the condition d . 12* (18.16) 180 ELECTBOD YNAMICS [Pajrt II In other words, the dimensions of the region in which the concept of a light ray is defined must be considerably larger than the wave¬ length of the light wave. For example, a small circle in the wall of a camera-obscura of diameter, say, 1 mm is considerably greater than the wavelength of visible light, which is of an order of magnitude 0.6 X 10-* cm. Therefore, the image obtained in a camera-obscura is formed with the aid of light rays. The optics of light rays is called geometrical optics. A ray is defined only when its direction is given, i.e., the normal to the wave front. If we are given a beam of nonparallel (for example, converging) rays, then the wave front is curved. But the radius of its curvature at each point is considerably greater than the wavelength. Such a converging beam of rays represents a set of normals to an “almost plane” wave. The curvature of the wave front close to the focus of the rays may become comparable with the wavelength, and then there arise devia¬ tions from geometrical optics. Such deviations are ealled diffraction effects. They are also observed when a light wave falls on some opaque obstacle. In accordance with geometrical optics, we should have ob¬ tained a sharp shadow—a transition from a region where the field differs from zero to a region where it is equal to zero. But Maxwell’s equa¬ tions do not permit such solutions, which are discontinuous in free space (cf. the boundary conditions, exercise 1, Sec. 16). In actual fact, there always exists a transition zone between “light” and “shadow,” in which the wave amplitude changes in a complicated oscillatory way. Exercises 1) Find the limiting dimensions of an object which may be observed in a microscope using light of wavelength X. Denoting the semiangle of the cone of rays, drawn from the microscope objective to the object, by 0, we have Ak = k sin 6. Whence Ax — ^ Afc A; sin 6 sinB It is therefore convenient to use a beam of rays with large solid angles and small wavelengths. 2) Show that if the dispersion law of exercise 19, Sec. 16, is used, then vl, the inequality can be seen directly. When e < 1 we have Sec. 19] THE EMISSION OF ELEOTBOMAGNBTIC WAVES 181 e + = 1 + 2 Sm * ' — ’ Squaring both sides of the inequality, it is easy to see that this quantity is greater than + = V7. See. 19. The Emission of Electromagnetic Waves Basic equations and boundary conditions. So far we have considered electromagnetic waves irrespective of the charges producing them. In this section we shall consider the emission of waves by point charges moving in a vacuum. The basic system of equations in this case is (12.37) and (12.38) together with the Lorentz condition (12.36). We revTite these equations anew: AA (19.1) A — oo, /•—>-oo) = 0, (19.4) A(<—> — oo , r— >-oo) = 0. If no boundary conditions are imposed on the solution of an inhomo¬ geneous equation, then any solution of the homogeneous equation can always be added to it so that a miique answer cannot be obtained. The radiation of a small element of charge. Let us begin with aninfinitesi- mal charge element 8e—pdV. We place it at the coordinate origin. Then 182 ELECTROD YNAMICS [Part II the solution of (19.2) will possess spherical symmetry. In Sec. 11 , an ex¬ pression for the Laplacian operator A was derived in spherical coordi¬ nates (11.46). As in the case of a static charge [equation (14.7)], we must retain only the term involving differentiation with respect to r and, this time, obviously, we must also differentiate with respect to time. For the time being we consider that the charge density at all points, except the origin, is equal to zero. Therefore, for all points for which equation (19.20) is written thus: JLAr* ^ _ — 0 r 2 8r dr c» dt^ ~ (19.6) Temporarily, we put (p = . Then 8 ? _ J. _^2 h. _ y. ® dr r dr r® ’ dr dr ’ dr ^ dr ^ dr^ ^ dr dr ^ dr^ (19.6) Substituting this in (19.5) and multiplying by r (by convention, r is not equal to zero), we obtain 8^0 _ 1 d^ dr^ (19.7) But this is the equation, of the form (17.6), for the propagation of a wave. Its solution is similar to (17.12): (D = i depends on the argument and the solution $2 depends on the argument t —y. The first of these arguments, t + —, for r->cx), — oo has a completely indeterminate form oo— oo, i.e., it is equal to anything. From the eondition (19.4), the funetion becomes zero when r->oo, t ->— oo. Therefore becomes zero for any value of the argument, i.e., it is equal to zero everywhere. (The potential at infinity must tend to zero more strongly than ~ so that there should be no radiation; see below in this section.) For the function Og, condition (19.4) denotes that 2 (— oo)=0. In other words, the function g tends to zero at minus infinity. It does not follow from this, of course, that it is equal to zero everywhere. Thus, Omitting the index 2 , we write the expression for 9 as follows: Sec. 19] THE EMISSION OP KLECTBOM.-VONBTIC WAVES 183 9 = (19.9) The function is not as yet determined. From the form of its argu¬ ment we conclude that it describes a travelling wave in the direction of increasing radii (because f>0). Such a wave is termed diverging. Retarded potential. The value of the function at j-= 0, < = 0 is shifted to the point r in a time < = -- or, in other words, the potential at the point r and time f is determined by the charge, not at the instant of time (, but at an earlier instant t——. The term is a measure of the 0 c retardation occurring as a result of the finite velocity of propagation of the wave. But when the retardation becomes a very small quantity, the potential very close to the charge must be determined by the instan¬ taneous value of the charge Se (t). We know from See. 14 that the potential due to a point charge is equal to (14.8), whence 0(t) ^ Se(t} ^ p(t)dV r r r ' Therefore, ^{t)==p(t)dV, (19.10) ( f* t —~ with (19.9) and (19.10), equal to is, in accordance (19.11) Now, displacing the coordinate origin to another point, we obtain, like (14.9), |E-r| dV. (19.12) Here it is assumed that the charge density is given at the point r (x, y, z), and the potential is calculated at the point R (X, Y, Z), thus introducing the explicit dependence of p on the spatial argument r. Finally, in order to obtain the complete solution to (19.2), we must integrate (19.12) over all the volume elements, i.e., with respect to d V—dx dy dz: |B-r| c (19.13) For point charges, p denotes the special function that was defined in Sec. 12. 184 KLKCTKODYIfAMICS [Part II Equation (19.1) has exactly the same form as (19.2) and its solution satisfies the same boundary conditions. Therefore, the vector poten¬ tial is written quite analogously to (19.13): A -J - c|il-rT (19.14) Comparing (19.14) with (15.10), which gives A for a stationary current, we see that J depends on the argument r in two ways: first, directly, in accordance with its spatial distribution and, secondly, via the time argument; since the system of currents is not infinitely small, but has finite dimensions, the retardation of a wave from various points of the system is different. Itolarded potential at a large distance from a system of charges. We shall now look for the form of the solutions of (19.12) and (19.13) at a great distance away from the radiating system. We note that the integrand depends on the argument i? in both integrals in two ways: in the denominator and via the argument t. The function in the denominator depends very smoothly on Jt. Its expansion in terms of powers of H yields terms which tend to zero like at large dis¬ tances away from the system. As will be shown later, they do not add anything to the radiation (for n>l). So we simply replace b'l'rge distances from the system, the term | R—r | , ai)pearing in the argument t of the numerator, looks like this; R-r|-R-rVR = K--^^f- =Ji-rn, (19.15) where n is a unit vector in the direction of B. The subsequent terms of the expansion (19.15) contain Jt in the denominator and are insignif¬ icant. 1’hus, at a large distance from the radiating system, the poten¬ tials are: (‘"■‘’I An estimate of the retardation inside a system. The term in the arguments of the integrands of (19.16) and (19.17) indicates by how much an electromagnetic wave, coming from the more distant parts of the radiating system, is retarded in comparison with a wave radiated by the nearer parts of the system. In other words, the term determines the time that the electromagnetic wave takes in passing through the system of charges. If the velocity of the charges Sec. 19] THE EMISSION OF EEECTKOMAGNETIO WAVES 185 is equal to v, then, in that time, they are displaced through a distance V The retardation inside the system is neghgible when this distance is small in comparison with the size of the system r. Therefore, if y r (or, more simply, v<^c), then the charges do not have time to change their' positions noticeably during the time of propa¬ gation of the wave in the system. However, in order that nothing should really change in the system, the eharges must also maintain their veloeities in that time, because the vector potential depends on the currents, i.e., on the particle velocities. This imposes a further condition which is formulated in the following manner. Let the charges oscillate and radiate light of fre- quency co. The wavelength of the light is equal to X = ——. In the time the phase of the charge oscillations changes by co ~. This change must be small in comparison with 2 ti, whence it follows that the size of the system must be small compared with the wavelength of the radiated light in order that the retardation inside the system should be insignificant. Thus, the terra in the argument of the integrand is small provided two inequalities are fulfilled: v-^c, r<|X. The vector potential to a dipole approximation. Let us assume that both the inequalities obtained have been fulfilled. We omit the term in the time argument in the expression for vector potential (19.17). Then the whole integrand will refer to the same instant of time t — — and we obtain C Recall now that i = pv and that the charges are point charges. Then the integral (19.18) is reduced to a summation over separate charges: i c R Here, the lower index t -denotes that the whole sum must be taken at that instant of time. But v' = —rr, so that at ’ , , , . d{«-—I f e Here we have used the definition for dipole moment (14.20). We note that (19.20) involves only a time derivative d. Therefore, the 186 ELEOTBOD YNAMICS [Part II transformation (14.21), which corresponds to a change of coordinate origin, does not change d either for a charged system or for a neutral system. In particular, (19.20) holds also for a single charge. The apiiroximation (19.20), in which A is expressed in terms of a derivative of the dipole moment of the system as a whole, is termed a dipole approximation. The Lorentz condition to a dipole approximation. In Sec. 17, a poten¬ tial gauge transformation for travelling plane waves was chosen such that the scalar potential became zero. We shall make the same gauge transformation for diverging spherical waves. To do this we must subject the vector potential to the following condition: divA = 0, (19.21) which is obtained from (19.3) if we take

which interrelate the coordinate and time, must change sign together with the velocity V. Otherwise, if the x and x' axes are turned in the opposite direction the equations will not preserve their form, and this is impermissible. Thus, the equations for the inverse transformation from unprimed quantities to primed have the same form as (20.2) and (20.3): X — ctx' — |3<', < = — yx' + 8t'. Comparing (20.7) and (20.5), we obtain — 8 * aS — PY ’ -P_. ^ aS — PY (20.7) ( 20 . 8 ) (20.9) ( 20 . 10 ) 13* 196 ELECTBOD YU AMICS [Part II From (20.10) it follows that a8-PY=l- (20.11) Then, from (20.9), we obtain a-8. (20.12) No other relationships are obtained from the comparison of the direct equations with the inverse equations. We now use condition (4). We divide equation (20.2) by (20.3): x' ~V “T + P (20.13) Let a: be a point occupied by a light signal emitted from the origin of the unprimed system at an initial instant of time <==0. Obviously, ~ =c. But in accordance with condition (4),-^ =c. Hence, * r «C + P YC + 8 ■ (20.14) We substitute the relations (20.4) and (20.12) into (20.14) in order to eliminate p and 8. There remains a relation between a and y: yc* + ac = ac — ocV , whence y=-a-^. (20.16) We now substitute an equation for a: (20.15), (20.4) and (20.12) into (20.11) and obtain (20.16) In extracting the square root, we must take the positive sign in accordance with condition (3), because then (20.3) becomes t'—t for a small relative velocity. A minus sign would yield t' — — t, which is meaningless. Now expressing ail the coefficients a, p, y, and 8 in accordance with equations (20.16), (20.4), (20.15), and (20.12), respectively, and substituting (20.3) into (20.2), we arrive at the required trans¬ formations : X — Vt (20.17) Vx t’ = lA 72 (20.18) Sec. 20] THE THEORY OE BBLATIVrry 197 In order to explain the meaning of these equations we shall apply them to some special cases. Let a clock be situated at the origin =0 of the primed system. It indicates a time t’. Then, from equation (20.20), it follows that ( 20 . 21 ) The clock which is at rest relative to its reference system we call the observer’s clock. It can be seen from (20.21) that one observer, comparing his clock with that of another observer, will always observe that the latter clock is slow, i.e., that If a clock is situated at the origin of the unprimed system (i.e., at the point * = 0), the transformation equation to the primed system is of the same form, since, from (20.18), we now obtain This not only does not contradict (20.21), but expresses that very fact: a clock moving relative to an observer is slow compared with his own clock. In the theory of relativity, a single universal time does not exist as in Newtonian mechanics. It is better to say that the absolute time of Newtonian mechanics is, in actual fact, an approximation, correct only for small relative velocities between clocks. The absolute¬ ness of Newtonian time has sometimes given cause to regard it as an a priori, logical category independent of moving matter. At any rate, Newton, by accepting instantaneous action at a distance, naturally had to consider time as universal; if we formally put c = cx3 in (20.18), we obtain t' =t. The instantaneous transmission of signals would allow us to synchronize clocks in all inertial systems independently of their relative velocities. In Newtonian mechanics, gravitational forces played the part of such instantaneous signals. It is sometimes thought that, knowing the velocity of light c, we can introduce a correction into the readings of clocks in different inertial systems such that the rate of time will everywhere be the same. But it is precisely equation (20.21) that describes the relative 198 BLEOTEOD YNAMICS [Part II passage of time in both reference systems after a correction has been introduced for the finite time of propagation of fight. Time reduction, as has already been shown, is completely reciprocal. Consequently, it can in no way be accoimted for by any change, resulting from motion, in the properties of clocks. The time reduction effect is purely kinematical. It must also be added that in speaking about clocks we by no means necessarily have in mind clocks which have been made by human hands; any natural periodic process that gives a natural time scale will do as well, for example, the oscillations in a light wave. It is clear that the physical properties of a radiating atom cannot in the least depend upon the inertial system in which the atom is described. This is what gives us the right to assert that equation (20.21) refers to similar clocks. At the same time we must remember that it is impossible to define time without relation to some periodic process, i.e., irrespective of motion. Relativity and objectivity. The relativity of time by no means indicates a rejection of the objectivity of its measurement in any given system of reference. It is entirely of no consequence what observer is observing the clock. The relative character of time in inertial systems is the only thing that counts. We have all long since become accustomed to the relativity of the datum line of time measure¬ ment related to time zones (i.e., to the sphericity of the earth). The theory of relativity teaches us that the time scale is also relative. The fact that an objective concept may be relative can be seen from the following example. In the Middle Ages it was thought that even direction in space was absolute, and it was then thought impossible to imagine that the earth was spherical since it would then foUow that our antipodes would have to walk upside down! The concept of “up” or “down” was related not with the direction of a plumb-line at a given point on the globe, but with certain other categories characteristic of the ideology of the Middle Ages. The vertical directions in Moscow or Vladivostok form a substantia] angle between each other, but nobody nowadays would think of arguing about which of them is the more “vertical.” The concept of verticality is completely objective at every point on the globe, but is relative for dijBFerent points. In the same way, time is objective in each inertial system, but is relative between them. Contraction of the length scale. We shall now consider the question of the measurement of length. In order to find out the length of a moving body (“its scale”), we must simultaneously plot the co¬ ordinates of its ends in a fixed system. Obviously, a fixed observer has no fundamentally different means of measurement, for, otherwise, he would have to stop the motion of the scale (i.e., transfer it to his reference system). If the ends of the scale are fixed by a stationary Sec. 20] THE THEORY OF RELATIVITY 199 observer at one time* we must put «=0. From (20.17), there foDows an expression for the length of a moving scale A*' measured by a fixed observer: Ax ( 20 . 22 ) Like (20.21), this equation has a symmetrical inversion. If a ■‘moving” observer measures a “stationary” scale, we must put — 0; it then turns out that Ax = -:r- r-T — . We conclude from V c* (20.22) that a moving scale is shortened relative to a stationary observer. The contraction occurs in the direction of motion. Lorentz supposed that this scale compression does not appear to both inertial systems, but, for some unknown reason, occurs when the scale moves relative to the “ether.” Lorentz and others attempted in this way to explain the negative result of Michelson’s experiment. Yet the very symmetry of the direct and inverse Lorentz trans¬ formations (20.17)-(20.2b) (they were known before the advent of the theory of relativity), from which the contraction of length follows only as a special case, shows convincingly that there is no system at absolute rest relative to an “ether.” It may be noted that by the beginning of the twentieth century, the “ether,” which had been introduced by Huygens as a medium that transmitted light oscillations, remained in physics simply as a rudimentary concept. The discovery and confirmation of the electromagnetic nature of light made the h 5 rpothetical elastic medium quite superfluous (see Sec. 12). Only the theory of relativity disclosed the real meaning of the Lorentz transformations. But then, many concepts regarded as absolute in Newtonian mechanics, turned out to be related to the motion of inertial systems. The formula for addition of velocities. We shall now find an equation for the addition of velocities arising from the Lorentz transformations. Differentiating (20.17) and (20.18) and dividing one by the other, we obtain dx' dt' = = dt V dx * c» dt (20.23) 1 - * The idea of the simultaneity of two operations performed in the same coordinate system may be imiquely defined with the aid of light signals. Indeed, observers at rest relative to each other at a given distance can always check their time with the aid of light signals by introducing a constant correction for the known time of propagation. 200 EUECTBOD YNAMICS [Part II Noting that dy'—dy and dz'==dz, we have a transformation of the velocity components perpendicular to V: dy' dt' V'y = V dx c* dt Vvx c‘ = Vvx c‘ (20.24) For small velocities, (20.23) and (20.24) become the ordinary equations for addition of velocities. This can be seen if we let c tend to infinity, i.e., by putting -y =0- It is easy to see that if v = V vi + ty + — c , then likewise v' —c, i.e., the absolute value of the velocity of light does not change in passing from one inertial system to another. But the separate components of the velocity of light, which are less than c, may of course change; the direction of a light ray relative to different observers differs, since there is no absolute direction in space. Abberation of light. In this connection let us consider the phenomenon of the aberration of light. Astronomical aberration, or the defiection of light, consists in the fact that stars describe ellipses in the sky in the course of a year. Their origin is easy to explain: the velocity of the earth, in annual motion, combines differently with the velocity of the light emitted by the star (Fig. 31). If the velocity vector of the starlight relative to the sun is tJS then the resultant direction of the velocity, for one position of the earth, is ET^ and, in half a year’s time, El\. These directions are projected on different points of the celestial sphere so that in the course of a year a star describes a closed ellipse. In angular 1 V units, the semi major axis of the ellipse is always equal to — where V is the velocity of the earth. ^ = 20".25. We may ask the question: Why does not the velocity of light in Michelson’s experiment combine with the earth’s velocity but remains equal to c, while the phenomenon of aberration shows that velocities combine (we note that Michelson’s experiment was also performed with an extra-terrestrial source of light). The explanation is that in Michelson’s experiment it was the absolute value of the velocity of light c that was measured (from the path difference of the rays), while in the aberration of light there is a change in the direction of the velocity of light as a result of the combination of its components with the velocity of the earth. Considering that the velocity of light relative to the sun is perpendicular to the plane £ Fig. 31 Sec. 20] THE THEORY OF RELATIVITY 201 of the earth’s orbit, we must put v* = 0, Vy=c, Vz = 0 into (20.23) and (20.24). Then the components of the velocity of light relative to the earth are ' T7 ' ^ Vx = — V , Vy= cV I -^. And, in accordance with Michelson’s experiment, v'x^-\-Vy'^—c^. The direction of the projection of the velocity of light onto the plane of the earth’s orbit (ecliptic) is reversed in the course of half a year, which is the reason why aberration occurs. Similar equations are obtained in the more complicated case when the rays from the star are not perpendicular to the plane of the ecliptic. They coincide with the equations that follow from a simple F® addition of velocities if terms of the order are neglected. Before the theory of relativity was put forward, it was wrongly supposed that the aberration of light contradicted Michelson’s experi¬ ment. Fizeau’s experiment. Fizeau’s experiment, which determined the velocity of light in a moving medium, was also believed to contradict Michelson’s experiment. Fizeau’s method was this. A beam of light was divided into two parts using a half-silvered mirror (Fig. 32). These beams were passed through tubes with flowing water; one beam in the direction of flow and the other in the opposite direction. For comparison, the same beams were passed through tubes in which the water was at rest. By .subsequent reflections the beams once again combined and cancelled each other when the path difference between them was equal to an integral number of half wavelengths (i.e., when they were in opposite phase). Coherence between them was obtained due to the fact that they both came from the same source. In stiU water, the path difference was chosen so that the rays were reinforced, i.e., the phase difference was equal to an even number of half wavelengths. The path Fig. 32 difference in flowing water was varied. Since the frequency of the light and the tube lengths remained un¬ changed, the change in path length indicated a change in the velocity of light relative to the tubes. First of all, we note that the result of Fizeau’s experiment in no way contradicts the general ideas about the relativity of motion. A reference system fixed in flowing water is not equivalent to a system fixed in the tube, if we are studying the propagation of light in water. 202 ELECTROD YNAMICS [Part II Since the velocity of light in water is equal to > where v is the re¬ fractive index of the water, the general equation for the addition of velocities (20.23) shows that does not remain a constant quantity when passing to another coordinate system. At the same time, we cannot use the simple velocity-addition equation, because the de¬ nominator of equation (20.23) differs from unity by-^ (F is the veloc¬ ity of the water). Considering that V<4c and expanding the de¬ nominator in a series up to the linear term inclusive, we find the change in the velocity of light in moving water (see exercise 1): It was precisely this value, which differs from that given by the simple velocity-addition law, that was obtained by Fizeau. Since Michelson measured c and Fizeau measured —, there is no contra- V diction between them. Interval. Despite the fact that x and t are changed separately by the Lorentz transformations we can construct a quantity which remains invariant (unchanged). It is easy to verify that this property is possessed by the difference cV — x^. Indeed, e^t'^ + - .2 - + 2F»'t' , x'^+ y.2 ^ - -^ I - r or c2<2 - = cH'^ - a;'* =- . (20.25) The quantity a is called the interval between two events; that which occurred at the coordinate origin a: = 0 and initial time f = 0, and another event that occurred at the point x and time t. The word “event” may also be regarded in its most common everyday sense provided that its coordinates and time may be defined. If the first event is not related to the origin of the coordinate system and the initial instant, then s* = c2 ( 1, for example, 5* = cH^ - (20.29) We shall plot the values of ct and I corresponding to two definite events measmed in quite dijfferent inertial systems. No matter what values are ob¬ tained as a result of these measurements of ct and I, the interval s (20.29) between the events is the same. It follows that the locus of the points, for all possible spatial distances I and time intervals ct, is an equilateral hyperbola s^^cH^ — P. Two branches of the hyperbola are possible: one lies in the fast relative to the event that occurred &tt — 0,x ~- 0, while the other is completely in the future. It is easy to see that such a relationship inevitably results if the events are causally related. Let the events, in some reference system, be known to occur in the same place, for example, sowing and reaping. To this system there corresponds the point 0 (sowing) and the point A (reaping). But since aU points of the given branch of the h 3 perbola lie at t>0, the sowing in any reference system must occur earlier than reaping. We can also proceed from causally related events which in our coordinate system do not occur at one point of space, such as firing and hitting the target (they occur at one point in a system fixed in the bullet). To the system fixed in the bullet there now corresponds a vertical section OA in Fig. 33, while to our system there corresponds some inclined line dra^vn from the origin to a point on the same upper hyperbola. Thus, here too, the second event—that of hitting the target—occurs after the shot in any reference system. 204 ELEOTaOD YN AMICS [Part II K the velocity of the bullet (or any material particle) is v, then s^—cH^ — It necessarily follows from the inequality (20.29) that l=vt- Answer: cosa'^--^-. I -cos a c 3) Write down the equations for the Lorentz transformations for an arbitrary direction of the velocity V relative to a coordinate system. Sec. 21] BBI/ATIVISTIO DYNAMICS 211 rV r'V In our equations x - —p- , x' =—pr— . Tlie component perpendicular to the velocity is From (20.17) V(rV) pa l2L V TV V VJi'Vl pT- Vt jpa C*' Multiplying this equation by -zpr and adding equations, we obtain r'- pa ^ ^ V(rVl pa Fn-Va 4) Write down the Lorontz-transformation equations for the components of acceleration. 5) Show that the “hnu’-dimensional volume element” dx dy dz dt is invariant with respect to a Lorontz transformation. dr = da:'11-from (20.22) and dt — dt' from (20.21), whonco the statement follows. (i) A light beam is within a solid-angle element dfi. Show that Lorentz transformations leave the quantity w^dO invariant. Uso the rosiJt of o.xcrciso 2: dO. — — 2iTd cos h. Sec. 21. Relativistic Dynamics Action for a particle in the theory of relativity. The adjective rel- ativistic denotes invariance with respect to Lorentz transformations, which invariance satisfies the relativity principle. For example, Maxwell’s equations in free space are relativistic. In effect, the Lorontz transformation is derived from the require¬ ment that the equations of electrodynamics remain invariant. There¬ fore, the proof of the relativistic invariance of Maxwell’s equations, which proof will be given somewhat later in this section, is simply in the nature of a confirmation. The situation with mechanics is altogether different. Newtonian mechanics satisfies only the Galilean relativity principle, which holds for velocities small compared with c. Therefore, it is necessary to find equations of mechanics such that they will be invariant with respect to Lorentz transformations. In Sec. 10 it was shown how to develop a mechanics by proceeding from the principle of least action. And it was found possible to deter¬ mine the form of the Lagrangian of a free particle by proceeding from two basic assumptions [see equations (10.11)-(10.13)]: 14* 212 ELECTBOD YNAMICS [Part II 1) Action is invariant to Galilean transformations; 2) The Lagrangian of a free particle depends only on the absolute value of velocity; the velocity vector v cannot be involved in it because, in the absence of an external field, there are no distinguishable directions (in space) relative to which the vector v can te given. In relativistic mechanics the first condition is replaced by the invariance to a Lorentz transformation, while the second condition remains unchanged. Both conditions are satisfied by an action function of the form 1 I o ( 21 . 1 ) *S’=Jads=Jac /1 — ^ dt, where we have used the relationship (20.30) between ds and dt. Agree¬ ment with the first condition can be seen from the fact that action is expressed in terms of interval only, while agreement with the second condition is obvious. No other invariant quantities can be constructed from dl and dt except the interval, whence the uniqueness of the choice (21.1). The Lagrangian lor a free particle. In order to define the constant a, we examine the limiting form of (21.1) for a small particle velocity. If __ fl-|--l-^t- (21.2) From the definition of the Lagrangian (10.2) S = jLdt it follows that the Lagrangian is L = ac|/ '1. a.c — ■ ^ c2 (21.3) (21.4) The first term in (21.4) is constant and can be omitted as not appearing in Lagrange’s equation [see (10.8)]. The second term should be compared with the Lagrangian for a free particle in Newtonian mechanics: ( 21 . 6 ) Whence L = a = ic. (21.6) The meaning of m here is the mass of the particle measured in a coordinate system in which the particle is at rest (or infinitely near rest). Thus, by its very definition, the quantity m is relativistically invariant. Finally, we have the Lagrangian in the form L = — mc^ L/1 (21.7) Sec. 21] RELATIVISTIC DYNAMICS 213 Momentum in relativistic mechanics. From (21.17), we immediately obtain an expression for momentum in the theory of relativity: 8L my ( 21 . 8 ) As required, at small particle velocities it reduces to the Newtonian expression p = mv. Sometimes the quantity ——- (i.e., the proportionality factor V1 — v^jc^ between velocity and momentum) is called the maas of motion of the particle, as opposed to the rest mass m. To avoid confusion we will not use the expression “mass of motion,” and will take the term mass to mean the quantity m which is relativistically invariant by definition. The limiting nature of the velocity of light. The limiting character of the velocity of light, about which we have already spoken in Sec. 20, can be seen from equation (21.8). As the velocity of a particle approaches the velocity of light, its momentum tends to infinity. The only exception is a particle whose mass is equal to zero. Its momentum, written in the form (21.8), gives the indeterminate form 0/0 for v—c and can remain finite. But then the velocity of this particle must always equal c. This property, as we know, is relativist¬ ically invariant since the velocity of light is the same in all inertial systems. The momentum of such a particle must be given in¬ dependently of its velocity [and not according to equation (21.8)], since the velocity is already determined and is equal to c. A velocity greater than c is utterly meaningless because it involves an imaginary quantity for momentum. Energy in the theory of relativity. Let us now determine the energy of a particle. In accordance with the general definition for energy (4.4), 8L j- ■■ V ^- L = dy ' [^1 r c + mc2 y 1 y> (21.9) Equation (21.9) once again confirms the limiting nature of the veloc¬ ity of light. When v tends to c, the energy of the particle S tends to infinity. In other words, an infinitely large quantity of work must be performed in order to impart to the particle a velocity equal to that of light. Rost energy. From equation (21.9), the energy of a particle at rest is equal to mc^. Let us apply this equation to a complex particle ca¬ pable of spontaneously decaying into two or three particles. Many atomic nuclei and also unstable particles (mesons) are capable of such disintegration. In the disintegration, the energy must be conserved, 214 ELBCTltOD Y NAMICS LPart II ^ = (21.10) because disintegration is spontaneous, caused not by any external interaction, but by some internal motion in the complex particle. The Lagrangian for this motion is not known explicitly, but in any case it cannot involve time. Therefore, the energy of a complex particle before disintegration is equal to the energy of the two particles formed after the disintegration, when there is no longer any interaction between them. The energy of all these particles is expressed m accordance with equation (21.9), as aj)plied to all free particles (whether simple or complex) when their motion is considered as a whole. The only possible form of the Lagrangian for such motion is (21.7), from which it follows that the energy is in the form (21.9). Substituting this expression in (21.10) and noting that the initial particle was at rest, we obtain TOC* = ( 21 . 11 ) But the terms S’l and on the right are correspondingly greater than TOjC* and TOgC*, whence wo obtain the fundamental inequahty TO ^ TOi + TO2 . ( 21 . 12 ) Hence the mass of a complex particle capable of spontaneous dis¬ integration is greater than the sum of the masses of its component particles. In Newtonian mechanics, the mass characterizing the motion of the system as a whole [see the last term of equation (4.17)] is equal to the sum of the masses of the component particles. If we define the difference T= (21.13) as the kinetic energy of a particle (for small energies it reduces to T = and call toc* the rest energy, then it can be seen from the law of conservation of energy (21.11) that part of the rest energy of a complex particle is converted into kinetic energy of the component particles, and part is converted into their rest energy. Only the total energies S, and not the kinetic energies T, satisfy the conservation law because the kinetic energy of a complex particle as a whole is equal to zero before disintegration and cannot be equal to the essen¬ tially positive kinetic energy of the disintegration products. In chemical reactions, the change in the rest masses of the reacting substances occurs in the order of 10“® (and less) of the total mass. Sec. 21] BELATIVISTIO DYNAMICS 215 In nuclear reactions, where the particle velocities are of the order c/10, the change in mass may approach one per cent. When an electron and positron (a positive electron) are annihilated, their energy, including rest energy, is totally converted into the energy of electromagnetic raidiation. As we shall see from quantum theory, radiation is propagated in space in the form of separate particles—so-called light quanta (quan¬ tum mechanics teaches that this is compatible with the wave proper¬ ties of radiation!). The velocity of a light quantum is equal to c so that its mass is identically equal to zero. For this reason, the total rest mass of the particles taking part in the annihilation process is 2 mc^ before the annihilation and zero afterwards. However, the change in the energy of the electromagnetic field is, of course, equal to 2 me*, provided the electron and positron did not have any additional kinetic energy. We could, by convention, call the energy of an electromagnetic field, divided by c*, its mass. With such a definition of mass, the total “mass” would be conserved. But com- ])ared with the law of conservation of energy, such a law of conserva¬ tion of “mass” does not contain anytliing new; it only repeats the law of conservation of energy in other units. It is precisely the rest mass that is best to use in describing nuclear reactions, for a change in rest mass determines the energy which may be generated as a result of the reaction (in the form of kinetic energy of the disintegration products, or in the form of radiated energy). There is no sense in calling the energy of a light quantum divided by the square of the velocity of light, its mass, because this quantity does not in any way characterize light quanta. Tiiis quantity has one value in one reference frame and another value in another frame, because the energy of any particle depends upon the reference system relative to which its motion is defined. Yet rest mass is a quantity that characterizes the particle. For example, the rest mass of an elec¬ tron, involved in the expressions for all its mechanical integrals of motion, is equal to 9 x 10-*® gm. The corresponding quantity for a quantum is identically equal to zero, and, in this sense, characterizes a light quantum in the same way that the quantity 9 x 10-*® gm is characteristic of an electron. The mass of a particle determines the relationship between the momentum and velocity of the particle in accordance with equation (21.8). It is impossible to determine the mass of a particle by its momentum alone, since particles with the same momenta can have quite different masses. For this reason, it is meaningless to state (though this is sometimes done) that the existence of light pressure (i.e., momentum of the electromagnetic field) proves that the fight quantum has a finite mass. It is sometimes said that a mass of one gramme is capable of releasing an energy of 9 x 10*® ergs (i.e., 1 c*). However, if the substance con- 216 EUECTROD YNAMICS [Part II sists of atoms the possibility of generating this energy is still question¬ able since up to now not a single process is known in which the total quantity of protons and neutrons (collectively called, nucleons) is changed.* This is why, the relative change in rest mass in nuclear reactions is always measured in fractions of one percent. The possibilities of various reactions are also limited by the conser¬ vation of total charge. The Hamiltonian for a free particle. We shall now express energy in terms of momentum. Squaring equation (21.9) and subtracting from it equation (21.8), after it has been squared and multiplied by c^, we obtain — — (21.14) We have called the energy expressed in terms of momentum the Hamiltonian [see (10.15)]. Hence, S — = Vm^c* + . (21.15) Whence we obtain a relationship between the energy and momentum of a particle that has no rest mass: S = cp. (21.16) The Lorentz transformation for momentum and energy. We shall now find out how energy and momentum behave with respect to a Lorentz transformation. From equation (21.8) we get Px- mvx m dx f dt F4 = mc dy dz Px = mc-^ mc^ dt da dt dt /I v‘ da • (21.17) The quantities m, c and ds are invariant. Hence, the components Px, Py and px are transformed similar to dx, dy, and dz, i.e., similar to X, y, and z. In accordance with the last equation, energy transforms like time. We can make the following comparison; x, y, Pz~ z, ^ r^cH. * In order to annihilate the whole mass we would have to first prepare “antimatter” (when ordinary matter interacts with antimatter they are mutually annihilated, cf. Sec 38). But this would require a like expenditiuo of energy. See. 21] RELATIVISTIC DYKAMICS 217 Now mation substituting momentum and energy in (20.17) and (20.18), we obtain the Lorentz transfor- (21.18) Vy = Py> (21.19) Pz=^Pz, (21.20) 11 (21.21) We note that a correct transition from (21.18) to a nonrelativistic equation for the transformation of energy is obtained only when the rest energy mc^ is substituted in place of for then = —mV (i.e., v'x = Vx — V) in agreement with the Galilean law for addition of velocities. Hence, if we demand that the Lorentz transformation yield the correct limiting transition to a Galilean transformation, it is necessary to include the rest energy of the particles in their total energy. Con¬ versely, the kinetic energy T (21.13) does not give a correct limiting transition. Further, we note that if we form the expression from equa¬ tions (21.17) we obtain ^2 _ in accordance with (21.14) The velocity of a system of particles in the theory of relativity. We shall now show how to determine the velocity of a system of particles in relativity theory. We shall consider two particles. Between the veloc¬ ity, momentum, and energy of each particle there exists the relation P = 4-- (21.22) It is obtained if we divide (21.8) by (21.9). The same equation can also be obtained somewhat differently. Let us determine, from (21.18), the velocity F of the coordinate system relative to which the momen¬ tum of the particle is equal to zero. Putting = 0 on the left-hand side of (21.18) we will have, on the right, T7 . PxC’‘ s or, if the velocity is not along the a;-axis at all, in accordance with (21.22) V=pc*/^=v. As applied to a single particle, the statement 218 BLECTBOD YNAMICS [Part II v=V is trivial and simply denotes that the momentum of the particle, relative to a coordinate system moving with the same velocity as the particle itself, is equal to zero. We now ai)ply equation (21.18) to two particles in order to find the velocity of the coordinate system relative to which their total momen¬ tum is equal to zci-o. The total momentum in the primed system is p'-fp'--p', and the total energy Let us take the a;-axis along p'. Since the Tjorentz transformation is linear and homogeneous, it has the same form for the sum of two quantities as for each sei>arate- ly. Therefore, we immediately obtain an equation similar to (21.22): V _ (ill '--Pil (21.23) I’lie primes may bo omitted here. In order to obtain the limiting tran¬ sition to the velocity of the centre of mass in Newtonian mechanics from (21.23), it is necessary to take Pi=-»«jVi, p 2 =wi.jV 2 , ^2 C-, i.e., the particle energies are replaced by their rest energies. 'I'lie quantity V, expressed in terms of the particle velocities accord¬ ing to equations (21.8) and (21.!J), does not have the form of a total derivative of any quantity with respect to time. Therefore, it is im- possibl(! in relativistic mechanics to determine its coordinates in terms of tlie velocity of the centre of mass. It is better to say that if we attenqit to express the coordinates of the centre of mass by means of a classical (or some other) equation it is impossible to represent V in tlie form of a time derivative of these coordinates, except in the trivial case when and V 2 are constant. This is \vh 3 ?^ the concept of centre of mass for particles moving in accelerated motion cannot be used. As regards relative velocity, v,^- -Vo, it is meaningless in relativistic mechanics, since there is Jio simple law for the combination of veloc¬ ities. Action for particles in an electromagnetic field. Let us now turn to the equatioirs of motion for a charged particle in an electromagnetic field. We already know tlie part of the action function which describes the interactioii of cluarges and field. Jfrom equation (13.17), this is S^. Since the variation of (S'j leads to Maxwell’s equeations, we can be cer¬ tain of the relativistic invariance of i6\. As applied to point charges, we have already written in magnetostatics in equation (15.26). But action for free cluarges in the next equation, (16.27), was suitable only for small particle velocities. We now know the Lagrangian for a fast particle in the absence of a field (21.7). Thus, the Lagrangian in an external field is equal to the sum of the rclativistically invariant expressions (21.7) and (16.26): P = + — e on the left-hand side what dt 8 Pq d/t di dfv we have is the required quantity for the change of kinetic energy in unit time. On the right-hand side the term v [v H] = [vv] H = 0 and there remains only the work done by the electric force: As was to be expected, the magnetic force [vH] does not per¬ form work on the charge because it is perpendicular to the charge velocity at every given instant of time. The Lorentz transformation for the field components. From (21.37) and equation (21.36), if we write it in terras of components, it is easy to obtain the Lorentz transformation equations for the field components. These equations must be written so that their form does not change in passing from one coordinate system to another. Let us take equation (21.36) for the component of momentum on .T, and multiply it by~. We shall also multiply equation (21.37) 17 by -y— and also by , where F is the relative velocity of the coordinate as c system. After this we subtract (21.37) from (21.36). Then on the left- hand side we have 222 ELECTROD YNAMICS [Part II On the right-hand side we will have the expression = « ^.(£-1- S-) + t(«- T t - + T • But ds is invariant. Therefore, the quantities on the right-hand side must be transformed in accordance with the basic equations (20.17) and (20.18). Differentiating these equations, we obtain ds ds ds [/ c* ’ ds ds ’ ds ds Now, dividing both sides of the equation by 1 —^ and miilti- plying by , we will have an equation for the a;-component of momentum m the new coordinate system: ') e V K + In accordance mth the ])rinciple of relativity, this equation must be written in the same way as for the unprimed coordinate system: dt' dt’ c. dt' ' Gomparhrg the last two equations, we obtain the field transformation equations: (21.38) V //y + — ' c (21.39) (21.40) In the same way, though from other equations (21.36), is it easy to find other equations for field transformation: (21.41) (21.42) Sec. 21] BEIiATlVtSTIO DYNAMICS 223 (21.43) Consequently, in contrast to coordinates, it is not the longitudinal but the transverse components that are transformed in the field. The change in field, in passing from one coordhiate system to another, is verified to a nonrelativistic approximation (i.e., to the ac- curacy of terms of the order -1 in a unipolar induction experiment. A diagram of tlie ex¬ periment is shown in Fig. 34. The magnet NS rotates around its longitudinal axis. Two col¬ lectors connected by a fixed conductor are joined to the centre of the magnet and to its axis. When the magnet is rotated, an e.m.f. appears in the wire. This experiment is frequently inter¬ preted as meaning that when the magnet is rotated the wire “cuts” its lines of force as if the lines were attached to the magnet like 34 brushes. Actually, unipolar induction must be understood as follows. There is only a magnetic field H in the coordinate system attached to the magnet, while the electric field is equal to zero. Hence, in a system fixed in the wire, relative to whicli the magnet moves, an electric field, too, should be observed in accordance with (21.42) or (21.43). This field is of an order of magnitude y- H and produces the e.m.f. We note that a coordinate system is defined only when both the electric and magnetic fields are specified. It is insufficient to specifiy only one of them. The invariants of an electromagnetic field. From equations (21.38)- (21.43), it is easy to obtain the following two invariants: = £2 - , (21.44) E'H' = EH. (21.45) From the invariance of these expressions it follows that the electro¬ magnetic field of a plane wave appears similar in all systems. Indeed, in a plane wave, E — H or — H^ = 0. This property is invariant according to (21.44). Further, E J. II, so that (EH) =0. This property is invariant according to (21.45). The quantity (E H) is invariant with respect to a Lorentz trans¬ formation. But with respect to a replacement of x, y, z by -x, -y, -z 224 ELECTKOO YNAMICS [Part II (i.e., an inversion of the signs of the coordinates), it is not invariant, because in this case E changes sign tvhile H does not (see Sec. 16). The quantity — 11^ is invariant even when the coordniate signs are inverted. But this quantity is the Lagrangian for a free electro¬ magnetic field. Integrated over the invariant volume dx, dy, dz, dt (exercise 5, Sec. 20), it yields invariant action, as required, while the quantity (EH) does not give a real invariant. The linearity of Maxwell’s equations with respect to field, A real invariant can be formed from the quantity EH merely by squaring. It is, of course, not at all obvious beforehand why such a quantity as well as the square of the invariant E^ — cannot appear in the Lagrangian for an electromagnetic field. Tlie same can be said of higher-order terms which do not change sign in the substitution of X by -X, etc. But if some terms—other than quadratic with respect to field—are left in the Lagrangian, then Maxwell’s equations will contain nonlinear terms. The essential difference between nonlinear and linear equations Ls that the sum of two solutions of a nonlinear equation is not its solution. Indeed, if two electromagnetic waves are propagated in a vacuum they are simply combined, and in no way distort each other. In nonlinear theory, the velocity is a function of the wave amplitude, while in electrodynamics the velocity of light is a universal constant. For this reason, the choice of the Lagrangian in the simplest form E^ — expresses the experimental fact that the law of variation of any electromagnetic field in space and in time is in no way dependent upon whether another field is operative in that same charge-free region of space. Actually the quantum electromagnetic field theory indicates the existence of certain nonlinear effects. In the range of phenomena for which classical electrodynamics is applicable, these effects are not essential. Transformation of charge density and current density. From the definition of charge density one can find the law of its transformation. Smee charge is an invariant quantity, we have de = pdxdydz = p^dx^dy^dz^, (21.46) where po is the charge density in a system relative to which it is at rest and, hence, the quantity is also invariant by definition. Whence, dyQ diZQ dt dt ^ dxdydz dt^ da ’ (21.47) where we have used (20.30) and exercise 5, Sec. 20. The current density is Sec. 21] RELATIVISTIC DYNAMICS 225 . dx dt dx dy dz Us • (21.48) From here it can be seen that the current components are trans¬ formed like coordinates, while charge density is transformed like time. Let us consider a conductor in which a current is flowing. In the coordinate system in which the conductor is at rest, it remains neutral, but in other systems a charge density must appear on it. This fact does not contradict the invariance of total charge, but follows from it in accordance with (21.46)-(21.48). The invariance ol action for the field. Let us now verify that the action term (13.17) describing the interaction of field and charge is invariant. It follows from (21.27) and (21.28) that the vector potential transforms like momentum (i.e., like a radius vector), while the scalar potential transforms like energy (i.e., like time). For this reason the product Aj - >„) and situated in a uniform magnetic field Hz = H, //* = ffy = 0. The oscillations of the charge are governed by the following nonrelativistic equations: m'x— — ma^x + — 'Hy, c my = —maly — ~Hx, mz— —mojjz. The third equation does not depend on the first two. The first two equations are easily solved if we put a: = £ie>“‘, y —bet'll. Then o(«g---—6 = 0, 7nc eH b (coq — «*) -f -a= 0. me Let us multiply the second equation by i and first subtract it from the first, and then add it. The combinations a±,ib then satisfy the equations eH (a ± ^ 6) (Wo — w^) (a ± » 6) *> =» 0, 15' 228 ELECTROD YNAMICS [Part II Cancelling a±ib, we arrive at the equations for frequencies eH by in the term » and represent the difference wo ■— a® (“ I" “o) (“—“o)> which is approximately equal to (a —o>o). Then we obtain expressions for the frequencies of both oscillations: ell e H They differ from the undisplaced frequency by, i.o., by the Larmor frequency ml- From the equations for a and 6, after substituting m — mj=F ml. we have, to the same degree of approximation. a = ±ib. If we represent the coordinates in real form, we obtain for both oscillations x — n cos (m„ ml) ti y= ±a sin (m„ T ml) t. Thus, the radius vector of a particle performing oscillations with frequency Oo + <^L rotates in a clockwise direction, while for oscillations with frequency Mo—ML it rotates in an anticlockwise direction. Thus, in accordance with Larmor’s theorem, the frequency ml is added to the frequency Mq or subtracted from it, depending upon the direction in which the charge rotates (we note that the sign of ml is changed for a negative charge). Let us consider the radiation of such a charge in a magnetic field. We know that the electric vector of the radiated electromagnetic wave lies in the same plane as the charge displacement vector. If radiation is observed to bo due to the 2-component of the dipole moment, its electric vector is along the 2-axis and is proportional to 2. Thus, the radiation is plane-polarized and is of frequency M^. The oscillation occurring along the field and having an undisplaced frequency radiates electromagnetic waves, which are polarized in the same plane as the magnetic field. This oscillation does not radiate at all in the direction of the magnetic field, but the oscillations with frequency m„ “L radiate circularly- polarized waves, and the electric-field vector rotates in the same direction as the charge displacement vector. All three frequencies radiate in a direction perpendicular to the field. How¬ ever, since the charge oscillations are viewed from one side in this position in circular rotation, the vector of electric-field oscillations lies in a plane per- pendicidar to the constant external magnetic field, so that waves with fre¬ quencies Mj it ML are now also plane-polarized. In observations that are not at right angles to the field, we obtain elliptically-polarized oscillations and a plane-polarized oscillation of frequency The calculations set out here form the classical theory of the Zeeman effect. The line splitting that is actually observed for various values of magnetic field is correctly described only by quantum theory (Sec. 34 ). PART III QUANTUM MECHANICS Sec. 22. The Inadequacy of Classical Mechanics. The Analogy Between Mechanics and Geometrical Optics The instability of the atom according to the classical view. Ruther¬ ford’s experiments in 1910 established that the atom consists of light negative electrons and a heavy positive nucleus of dimensions very small compared to the atom itself (see Sec. 6). For such a system to be stable, it is necessary that the electrons should revolve around the nucleus like planets about the sun, for unlike charges at rest would come together. This stability condition of the atom is, nevertheless, insufficient. In the case of motion in an orbit, electrons will experience centri¬ petal acceleration, but, as was shown in See. 19, a charged particle undergoing acceleration radiates electromagnetic waves, thereby transmitting its energy to the electromagnetic field. Thus, the energy of an electron moving around a nucleus should continuously diminish until the electron falls onto the nucleus. This statement is in striking contradiction to the obvious fact of the stability of atoms. The Bohr theory. In 1913, N. Bohr suggested a compromise as a way out of this difficulty. According to Bohr, an atom has stable orbits such that an electron moving in them does not radiate electro¬ magnetic waves. But in making a transition from an orbit of higher energy to one with lower energy, an electron radiates; the frequency of this radiation is related to the difference between the energies of the electron in these two orbits by the equation A <0 == S 2 , where A is a universal constant equal to 1.064 x 10“*^ erg-sec. Both of Bohr’s principles were in the nature of postulates. But it was possible with their aid to explain, in excellent agreement with experiment, the observed spectrum of the hydrogen atom and also the spectra of a series of atoms and ions similar to the hydrogen atom 230 QUANTUM MECHANICS [Part III (for example, the positive helium ion, which consists of a nucleus and one electron). Despite the fact that both of these, essentially quantum, postulates of Bohr were completely alien to classical physics and could in no way be explained on the basis of classical concepts, they represented an extraordinary step forward in the theory of the atom. Indeed, the first postulate contains the statement that not every state of the atom is stationary, but only certain states. This statement, as we now know, derives from quantum mechanics just as directly as elliptical planetary orbits derive from Newtonian mechanics. The Bohr theory was very successful m explaining the spectra of single-electron atoms. But the very next step, a two-electron atom such as the helium atom, did not yield to consistent calculation by the Bohr theory. The theory was even less capable of explaining the stability of the hydrogen molecule. For this reason, the situation in physics, notAvithstanding a number of brilliant results of the Bohr theory, was completely unsatisfactory. Besides the particular diffi¬ culties that we have noted here, the Bohr theory was, on the whole, eclectic, since it was inconsistent in its combination of classical and quantum concepts. Light quanta. The inadequacy of classical ideas appeared most obvious in the problem of the stability of the atom. But earlier there were many facts which classical (i.e., nonquantum) physics failed to explain. A case in point is the theory of an electromagnetic field in equilibrium with matter (for more detail, see Sec. 42). Here, classical theory leads to an absurd result—the total energy of an electromagnetic field in equilibrium with radiating matter is expressed in the form of a divergent, i.e., infinite, integral. In order to give a satisfactory description of experimental facts, Planck, m 1900, postulated that sources of radiation emit and absorb energy of the electromagnetic field in finite amounts. These discrete quantities, or quanta, as they were called by Planck, are proportional to the frequency of the emitted or absorbed radiation. It is easy to see that the factor of proportionality must be the same as in Bolir’s second postulate (actually, Planck introduced a quantity 2 n times greater, but used a frequency v = , equal to the number of oscil- lations per second). Bohr’s second postulate relates the properties of discreteness of stationary states of a radiating system (atom), occurring in line spectra, to the energy of the emitted quanta. Classi¬ cally, it is just as impossible to explain this discreteness as it is to explain Planck’s initial hypothesis. The duality of electro^namical concepts. At the beginning of the twentieth century, the classical theory of light also turned out to be incapable of explaining many facts without appeahng to an additional hypothesis concerning light quanta. But at the same time, Sec. 22] THE ANALOGY BETWEEN JMECHANICS AND GEOMETBIOAI, OPTICS 231 there was a whole range of phenomena, such as diffraction and inter¬ ference of light, which appeared to be intimately boimd up with its wave nature. It did not seem possible to explain these phenomena in terms of classical corpuscular concepts. Another group of phenomena could be explained only on the basis of Planck’s hypothesis concerning light quanta, and was in most obvious contradiction to the classical wave conceptions. Let us note two such phenomena. I refer, firstly, to the so-called photoemissive effect, i.e., the emission of electrons from the surface of a metal in a vacuum when the metal is illuminated by ultraviolet rays. The energy of the photoelectrons depends only on the radiation frequency and is independent of its intensity. This can only be understood if we assumed that the energy of electromagnetic radiation is absorbed in the form of quanta, Aco. Then the kinetic energy of an electron will be equal to the energy of the quantum minus the electronic work function (the energy needed to remove the electron from the metal). Einstein, to whom this explanation of the laws of the photoelectric effect belongs, went further apd assumed that electromagnetic radiation is not only absorbed and emitted in the form of quanta, but is also ‘propagaied in that form. Since the energy of a quantum is equal to Aw, and its velocity is c, it should possess a momentum (see Sec. 21). It follows that a quantum is a particle of zero mass. It will be noted that the energy and momentum of an electromagnetic wave are related m exactly this way (see Sec. 17). The second phenomenon which exhibited the quantum properties of radiation provided for confirmation of Einstein’s hypothesis concerning the momentum of light quanta. In the scattering of X-radiation by electrons, the latter may be regarded as free, since the characteristic frequencies of their motion in a substance are very small compared with the frequency of the incident radiation (we have considered such scattering in exercise 4, Sec. 19). It is essential that in accordance with classical theory the scattered radiation must be of the same frequency as the incident radiation. But exper¬ iment showed that the frequency of the scattered radiation is less than the frequency of the incident radiation, and depends upon the angle at which the scattering is observed (Compton effect). The displaced frequency can be calculated in relation to the scattering angle, if it is assumed that the act of scattering occurs as the col- lison of two free particles—a moving quantum and an electron at rest. A collision of this sort was considered in exercise 5, Sec. 21, especially for the case of an incident particle with zero mass. The equation obtained there gives a perfectly correct description of the frequency shift in the Compton effect, if we consider that the energy 232 QUANTUM MECHANICS [Part III of the quantum is equal to ^ = Ato and its momentum p = A k (or A 0 X As stated, the first term in the last expression must be discarded when X->0. Whence, dt^ — X 8« / ® X and similarly AE^-Eo(y VxJ^osy. Substituting these expressions into the wave equation (22.3), we obtain a first-order differential equation for the phase

27 c (23.3) [see (18.9)], is small in comparison with k. Substituting Afc* for an electron in equation (23.1), wo obtain the analogous expression of quantum theory: Apx'^x~2Tth**. (23.4) Tliis is the so-called uncertainty relation of quantum mechanics. The concept of an electron trajectory has reasonable meaning if the un¬ certainties of aU three momentum components Ap*, Apy, Apx are small compared with the momentum itself: Apx4: Px, Apy<4py, Apz< Pz. {23.5) It may be pointed out that we have all along been saying “electron” simply to be specific. The same applies to a proton, neutron, meson and the like. Let us suppose that the track of an electron in a cloud chamber is 0.01 cm wide and the electron energy is equal to 1.000 ev = 1.6 x 10"® 4. fi V 1 n-io erg (1 electron-volt = -= 1.6 x 10"^® erg). According to (23.4) * An electron passing through a gas ionizes the atoms in its path. The supersaturated vapour with which the chamber is filled condenses on the ions. Upon illumination, the droplets appear in the form of a cloud track. ** Sometimes, Ap*, Ax are meant to signify not the “spreadings” p* and x in themselves, but their mean square values. Then, Ap* Ax > h. Sec. 23] BLECTBON DIFFBACTION 243 the component of momentum perpendicular to the trajectory has an indeterminacy 6.6 X 10-« 10-a = 6.6 X 10-*6, and the momentum itself is p = V2^ = V2.9 X 10-«« = 1.7 X 10-1®. It follows that the relationships (23.5) are satisfied to an accuracy of up to four parts in ten million. The observation of a track in a cloud chamber does not allow us to determine the trajectory with accuracy sufficient to notice deviations (in electron motion) from the laws of classical mechanics. The limitations of the concepts of classical mechanics. Thus, quantum mechanics does not abolish classical mechanics, but contains it as a limiting case, much the same as wave optics includes the geometrical optics of light rays as a limiting case. As we shall see later, quantum mechanics is concerned Avith the same quantities as classical mechan¬ ics, i.e., energy, momentum, coordinates, moment. But the finiteness of the quantum of action h imposes a limitation on the applicability of any two classical concepts (for example, coordinates and momen¬ tum) for one and the same motion. The coordinate and momentum of an electron cannot simultaneously have precise values because the motion is wave-like. To attempt to define these precise values is just as meaningless physically as it is to seek precise trajectories for light rays in wave optics. In the same way that it is impossible to obtain, as a result of improvements in optical devices, a precise definition for light rays in wave optics, any progress in measuring techniques as applied to the electron will not allow us to determine its trajectory more precisely than indicated by the relation (23.4), since strictly speaking the trajectory does not exist. Attempts are sometimes made to interpret relation (23.4) errone¬ ously. It is taken that a trajectory cannot be determined because the precision of the initial conditions does not exceed Apx and A a; connected by relation (23.4). This would mean that some actual tra¬ jectory does exist but that it lies within a more or less narrow region of space and within a certain range of momenta. The “real” trajectory is likened to the imaginary trajectory from a gun to a target before firing. The path of the bullet is not precisely knovni beforehand, if only because strictly identical powder charges cannot be obtained. But this inaccuracy in the initial conditions for the bullet only leads to a smooth dispersion curve for the hits on the target, while the distri¬ bution of electrons indicates diffraction effects. The presence of dif¬ fraction shows that no “real-though-imknown-to-us” trajectory exists. As a matter of fact, relation (23.4) by no meams indicates with what error certain quantities may be measured simultaneously, but to what 16* 244 QUANTUM MECHANICS [Part III extent these quantities have precise meaning in the given motion. It is this that the uncertairdy principle of quantum mechanics expresses. The term “uncertainty” emphasizes the fact that what we are con¬ cerned with is not accidental errors of measurement or the imperfection of physical apparatus, but the fact of momentum and coordinate of a particle being actually nonexistent in the same state. Sec. 24. The Wave Equation The wave function. Diffraction of Mght occurs because the wave amplitudes are added. When the wave phases coincide the intensity (which is proportional to the square of the resultant amplitude) is maximum; when the phases are opposite the intensity is minimum. In the diffraction of electrons, a quantity similar to intensity is meas¬ ured by the blackening of a plate, which blackening is proportional to the number of electrons incident on unit area. The distribution of the blackened grains on the plate obeys the same law as in the case of the diffraction of X-rays (in the sense of alternation of maxima and their relative positions). Thus, in order to explain the diffraction of electrons we must assume that with their motion there can be associat¬ ed some wave function whose phase determines the diffraction pattern. At the end of this section, we will show in the general case that such a wave function must be complex, since a real wave function cannot correspond to just any type of motion. Probability density. Electrons move independently of one another and pass through a crystal singly, as it were. Therefore, the number of electrons in an element of volume dF is proportional to the proba¬ bility of the appearance of one electron. Probability (a quantity simi¬ lar to light intensity) must be quadratic with respect to the wave function, in the same way that light intensity is quadratic in wave amplitude. But since probability is a real quantity, it can only depend on the square of the modulus of the wave function. Let us put dw = \ ' fik dp (24.3) (24.4) Hence, it coincides with the particle velocity as it should in accordance with (22.12), thereby confirming (24.2). The relation between wave function and action. We note that the wave function (24.3) may be written m the form (|) = e (24.6) * Or at least from invariance to Galilean transformations. 246 QUANTUM MECHANICS [Part ELI where 8 is the action of the particle. Indeed, the action of a free par¬ ticle is S=- classical mechanics I wave optics i quantum mechanics The vertical arrows denote a transition from rays or trajectories to wave patterns, while the horizontal arrows denote a transition from waves to particles. The latter relates only to nonquantum electrodynamics because in the transition to quantum field equations the need arises for a corpuscular representation (see Sec. 27). Here, we consider only the analogy between quantum mechanics and classical wave optics. The range of application of various theories. The regions in which quantum mechanics and wave optics can be applied do not overlap anywhere; in wave optics or, what is just the same, in electrodynamics, the velocity of light c is regarded as finite but the quantum of action h is considered arbitrarily small. In nonrelativistic quantum mechanics, c is considered arbitrarily large while h has a finite value. A quantum theory of the electromagnetic field, in which both h and c have finite values (i.e., the velocity ranges are comparable with c, and quantities with the dimensions of action are comparable with h), has, in essentials, also been completed at the present time. At any rate, any concrete problem requiring the application of quantum electrodynamics, may be uniquely solved to any required degree of precision, and the results agree with experiment. The existence of a light quantum as an in¬ dependent particle is not a supplementary hjrpothesis which must be made in order to formulate quantum electrodynamical equations. The consistent quantization of electromagnetic field equations necessarily leads to the corpuscular aspect of the theory (for more detail see Sec. 27). Sec. 24] THE WAVE EQUATION 249 The nonrelativistic particle quantum mechanics (i.e., constructed on the relation S = + U) is, in the region for which it is applicable, a theory which is just as consummate as Newtonian mechanics. Like the equations of Newtonian mechanics, the wave equation (24.11) is valid only for particle velocities small compared with the velocity of light. But still, in the region for which it is applicable, it is just as firmly established (in the same sense) as are the Newtonian laws for the motion of macroscopic bodies. The grounds for this are absolutely the same—^both nonrelativistic quantum theory and Newtonian mechanics agree with the widest range of experimental data, never contradicting them and providing for correct and unique predictions. In addition, they nowhere contain contradictory statements. The latter condition is, of course, not sufficient for a physical theory to be correct but it is at least necessary. The Bohr theory, or the old quantum mechanics (as it is otherwise known) did not satisfy this requirement; in addition to the classical concept of trajectory, it involved the quantum concept of discreteness of states. For this reason, it had always been clear that the Bohr formulation of quantum theory was not final and should be revised, no matter how wide the range of experimental facts that it explained. Quantum mechanics permitted the construction of a consistent theory of atomic structure. The actual calculation of wave functions for electrons in complex atoms is a problem of enormous mathematical complexity.* However, it is, of course, by no means the purpose of quantum mechanics to calculate the spectra of complex atoms; the essential point is that quantum mechanics allows us to systematize atomic and molecular states in such a way that the very nature of the spectra is understood, whereas classical mechanics could not explain even the stability of the atom. Thanks to quantum mechanics such fundamentally important facts as the chemical affinity of atoms or Mendeleyev’s periodic law are now understood. In its domain, quantum mechanics will, of course, perfect methods of approach to various concrete problems. The correctness of its general principles will serve as a basis for such refinement. The normalization condition for a wave function. Let us return to the wave equation (24.11). We shall write it for a wave function t}* and a conjugate function ({/* (in the second equation we have to replace — i by i): h Si}' i dt i dt 2m 2m At};* + U<\i*. * It is considerably simplified thanks to approximation methods suggested by V. A. Fok. 250 QUANTUM MECHANICS [Part III Let US multiply the first equation by (}'*. and the second by tj(, and subtract the second from the first. The term C/ij; is eliminated and the remaining terms give The left-hand side of the last equality is transformed to the form -Thr^=-Tit\^V- We can write the right-hand side more fully thus: “ ('J'* A(j; - (j^A:];*) = - {>P* divgrad divgrad (J;*) = = —div (+Url, = Si>. (24.22) As we shall see in the next section, this equation has a solution (vhich does not satisfy definite necessary conditions for all values of S. Thus, it turns out that, in contrast to the energy in classical mechanics, the energy of a quantum system cannot always be arbitrarily given. Exercise Prove that if there are two solutions of (24.22) for different values of energy f and S', then J ^*(r,S)^(r,S')dV The functions (r, and ^ (r, ^') satisfy the equations —A4i -1- I7il( = S'^. 252 QUANTUM MECHANICS [Part III Let us multiply the first equation by i|j. the second by 4'*. and subtract the second from the first. Integrating over the whole volume, similar to (24.19), and then transforming the volume integral on the left to a surface integral, we obtain the equation J T’ = 0- It follows that if then the second factor is equal to zero as required. This is the so-called orthogonality property of wave functions. It will be shouii in more general form in Sec. 30 because it forms one of the most important principles of quantum mechanics. Sec. 25. Certain Problems of Quantum Mechanics In this section we shall obtain solutions to the wave equation for certain cases which are partly illustrative and partly auxiliary. Nevertheless, many important laws are explained from these examples. We have already obtained the solution of the wave equation for a free particle (24.3). We shall now examine the solutions for bound particles. A particle in a one-dimensional, infinitely deep potential well. Let us suppose that a particle is constrained to move in one dimension remaining in an interval of length a, so that 0 a. We put C/ = 0 at 0 G\a , = -= = The second term of the integrated expression becomes zero at both limits in accordance with (26.8). Thus, See. 26] OEBTAIN PBOBLEMS OF QUANTUM MECHANICS 255 = (25.12) Real wave functions. The wave function (26.12) is real. Therefore, from (24.20), the current in this state is equal to zero. This can also be seen in the following way. The wave function (25.12) can be ex¬ panded into the sum of two exponentials. Each such exponential represents, together with a time factor, the wave function of a free particle (24.3), one of them corresponding to a momentum 'p — hx and the other, to the same momentum but with opposite sign. Thus, a state with Avave function (25.12) is represented as the superposition of two states with opposite momenta, these states having equal amplitudes. The mean momentum for a particle moving in a potential well according to classical mechanics is equal to zero; it changes sign for eiwy reflection from the walls of the well. In this sense, we can say that the mean momentum is also equal to zero for quantum motion. The difference is that at every given instant classical mo¬ mentum possesses a definite value, while the quantum momentum of a particle in a well never has a definite value ; the wave function involves states with momenta of both signs. This corresponds to the uncertainty principle; since the particle coordinate is within the limits 0(x,a2,z) = <\i(x,y,as)=0. (25.13) The wave equation must now be written in three-dimensional form: 2m \ dx^ ‘ dy^ fiz* / ^' (26.14) It is convenient to write the solution as follows: ij< = (78inxia; • sinx^y • sinxgZ. (26.16) It is written only in terms of sines and not cosines so as to satisfy the first line of the boundary conditions (25.13). We substitute (26.15) 266 QUANTUM MECHANICS [Part III in (15.14) and, utilizing the fact that for every factor of (25.15) there exists an equality of the form, this gives sin sin x ; A 4- = - (xj + x“ + x|) . (25.16) To satisfy equation (25.14) the energy must involve x^, in the following way; 2wi (xj + X» + X* Xj and Xg (25.17) The quantities x^, Xg, Xg are determined from the second line of the boundary conditions (25.13). The factors of (25.15) convert to zero either at x — a^, or y — a,^ or z — a^. In other words. sinxiai = 0, Xicq = ni7t; sinxgttg^^’ = (25.18) sinx3a3 = 0, y.^a^ = n^v:. Here Wj, and % arc integers of which none are equal to zero (other¬ wise <4 would be equal to zero over aU the box). Substituting x^, Xg, Xg from (25.18) in (25.17), we have the energy eigenvalues tp _ , nl ^ ® ~ 2 m Uf ■ "I «§ (25.19) The least possible energy is ’111 — fh^lj 2m \ af (25.20) It follows that the value S’ — O is impossible. Calculating the number of possible states. To each value of the three numbers n^, and Wg, there corresponds a single particle state. Let the numbers n^, n^, and Wg be large in comparison with unity. Such numbers may be differentiated: the differential dn^ denotes a number interval which is small compared with Mj, but stiU including many separate integral values of Then it stands to reason that there are exactly dn^ possible integers, included within the interval dui (and similarly within the intervals dn^ and dn^). Let us plot Ui, Wg, and Wg on coordinate axes. In this space we construct an infinitely small parallelepiped of volume dn^ dn^ dn^. In accordance with the foregoing, there are dui dn^ d«g groups of three integers n^, Tig, Wg in this parallelepiped, each corresponding to one possible state of the particle in the box. Altogether, the number of such states in the examined interval of values Wj, Wg and Wg is dN (rii, Wg, Wg) = duidn^dn^ . ( 25 . 21 ) Sec. 25] CERTAIN PROBLEMS OP QUANTUM MECHANICS 257 Substituting here x^, x^ and Xg from (25.18), we obtain another ex¬ pression for the number of states: dN (xi,Xg,Xg) = ^ ^ (25.22) where F=ai Ug Ug is the volume of the box. The numbers x^, Xg and Xg take only positive values. It was pointed out above that to each value of x there correspond two values of the momentum projection, which are equal in magnitude and opposite in sign. Therefore, if we compare the number of states included in the intervals dx,, and then there are half ^ h h as many states for the latter. Correspondingly, the number of states in the interval of values of momentum dfx dpy dpz is dN (px, py, pz) = V dpxdpydpz (2it h)^ (26.23) whore px, Py and p^ assume all real values from — oo to cxj. Equation (25.23) agrees with the uncertainty relation (23.4). If the motion is bounded along x by the interval a^, then only those states differ physically for which the momentum projections differ by not less than . Hence, there are states within the interval dpx. Multiplying —arrive at (25.23). In order to ensure coincidence of numerical coefficients with the results of rigorous derivation from the wave equation when evaluating the number of states from the uncertainty relation the quantity 2-Kh was selected on the right-hand side of (23.4) or 2tz from (18.10). We shall now consider the number of states after changing somewhat the inde- jiendent variables. We plot the quantities Xj, Xg and Xg on the coordinate axes (Pig. 37). Let us construct in this “space” a sphere whose equation looks like + X -t- Xa = K* . The numbers Xj, Xg and Xg are posi¬ tive so that we shall bo interested only in one eighth of the sphere; this octant is shown in Fig. 37. How many states are included between the octants of two spheres with radii K and K-f-dK? The number of states is equal to the integral of (25.22) over the whole volume between the octants, or 17 - 0080 258 QUANTUM MECHANICS [Part III dN (K) ^ J dN (Xi, Xg, X3) V. 4:tK2dK 8n* FKi*dK 27c“ (25.24) This is evident simply from the fact that the volume is equal to the surface of the octant multiplied by dK. But from (25.17) K is very simply related to the energy of the particle: ^_ s/im^ Whence h dN ( 0 is everywhere equal to U^, except within a region of width a near the coordinate origin, which region we called the well. For a:<0 the potential energy is ~ infinite (see Fig. 38).* Since the solution will be of different analytical form inside and outside the well, we must find the conjugation conditions for the wave func¬ tion at the boundary. Let us take the wave equation. a Fig. 38 2m dx^ + C/4- (25.26) * It was shown in Sec. 19 that the three-dimensional wave equation can be reduced to a single-dimensional one, with the difference that the variable r must be positive by its very meaning. This may be attained formally by situating an infinitely high potential wall at r = 0. Fig. 38 actually refers to a spherical potential well with an angular-momentum value equal to zero, when there IS no “centrifugal” term in the potential energy [see (5.8) and (31.6)]. Sec. 25] CERTAIN PROBLEMS OP QUANTUM MECHANICS 259 ajid integrate both sides over a narrow region a —a+8, including the point of discontinuity of the potential energy x = a. The integration gives a+8 2 m r/ii\ _ 1- f L\ dx 1 a+8 \dx j a-8 . 1“ J a-S (26.27) Even though U sufiFers a discontinuity at the boundary of the well, on the right it remains everywhere finite by arrangement. Therefore, when 8 approaches zero, the integral on the right also approaches zero. It follows that the left-hand side of (25.27) is also zero. In other words, (26.28) the limit of the derivative on the right is equal to its limit on the left. This argument would not hold in the problem of an infinitely deep well because then the integral in (25.27) would be indeterminate. Besides we notice that the derivative is finite at the points a; = 8 simply because the only solutions of equation (25.26) are those with finite derivatives (exponential function, sine or cosine). We shall now show, by means of a limiting transition, that even the wave function itself does not suffer a discontinuity at the boundary. Let U initially have a finite discontinuity region of width 8 and let the discontinuity of the function be A. Before passing to the limit the derivative in the region of discontinuity is of the order of ~ , so that when 8-^0, it diverges. Let us now multiply both sides of (25.26) by t}' and perform a transformation by parts: Let us integrate the transformed expression between a — 8 and a-f^. We then obtain 4+6 4+8 4—8 4—8 (26.59) We shall now perform the foregoing limiting process by indirect proof. We may write the integrated terms thus: (t II) 1) *)i (1^) . because the derivative as was shown, is not subject to a dis¬ continuity. Within the assumed discontinuity region of the iJ;-function, 17* 260 QtTANTtrM MECHANICS [Part III is of the order of —, but at the boundaries of the region it reverts to values which are independent of 8 and are therefore finite in the limit. Hence, the whole integrated part on the left in (25.29) is of the order of A -^ • The remaining integral is estimated as a-S Hence it tends to infinity as 8 tends to zero. The right-hand side of (25.29) is finite for 8 ->- 0 . Thus, by assuming that <{' has a finite dis¬ continuity A we have arrived at a contradiction. It follows that i|; is continuous at the point a together with its first derivative. Solutions in two regions. The wave equation for the region 0 < o (inside the well) is of the form We take its solution 2m <{^=(71 sin yix , (25.30) where x is defined from (25.3). The solution involving the sine only is taken because at the left-hand edge of the weU, where the potential energy suffers an infinite discontinuity, satisfies the boundary condition (25.1): 9 (0) = 0. The wave equation outside the well, when x>a, is 2m dx^ (25.31) First of aU let us take the case S>Uq. Then, introducting the ab¬ breviated notation 2m ~hr (26.32) we obtain (25.31) in the standard form (26.4) whence da:* = - xf (p . sin Xi a: -f Gg cos Xj x . (25.33) We must now satisfy the boundary conditions on the right-hand edge of the potential well where U suffers only a finite discontinuity. According to these conditions the wave function is itself continuous, i.e., Gi sin xa = sin x^ a + Cg cos x^ a (25.34) and its derivative X cos xa = Xi Gg cos x^ a — x^ Gg sin x^a. (26.35) Sec. 25] CEBTAIN PROBLEMS OF QUANTUM MECHANICS 261 From these equations we can determine G^, and Cg in terms of C^, i. e., completely express the solution outside the well in terms of the solution inside the well. The equations (25.34) and (26.36) are linear with respect to Cg and Cg and have solutions for all values of coefficient. <^2 = xjsin>casinxja-|- xcosxocosxja xjsinxacos XjO — xeosxasinxja Cl, Ci. Therefore, the boundary conditions may be satisfied for any real values of X and Xj. Thus, Schrbdinger’s equation is solvable for all There is no discrete energy eigenvalues for > ZJg. We could adjust the potential energy in this problem to zero at infinity, i.e., consider it equal to zero for x > a and equal to — U for o> a;^ 0. Then the case which we have just considered would correspond to positive eigenvalues of the total energy. Now let S' < Uq. We introduce the quantity 2m ~hF (Uq—S) (25.36) The wave equation is now written differently from that for S > U^, namely Its solution is expressed in terms of the exponential function + C^e . (26.37) But the exponential e^x tends to infinity as x increases. For x = ooii would give an infinite probability for finding the particle, and no 00 finite value could be assigned to the integral J | (p dar . It follows that 0 a physically meaningful solution exists only for and must be of the form C.e- (26.38) Let us again try to satisfy the boundary conditions at x—a. This time they appear as follows: Cjsin xa = , x(/iCosxa X Cj e-*". (26.39) (25.40) Let us divide equation (26.40) by (25.39) in order to eliminate and Og. We then obtain X cot xa = (26.41) 262 QUAKTtTM MECHANICS [Part III From this equation we find the expression for sin xa: sm y.a = ± —====r= ± Vl + cot'*xa -W. (25.42) Let us reduce this equation to a more convenient form. From (26.3) a's/2m so that sin xa = ± • (25.43) only those solutions should be chosen for which ctg xa is negative, in accordance with (25.41), i.e., xa must lie in the second, fourth, sixth, eighth, etc., quadrants. We shall solve this equation graphically (Fig. 39). The left-hand side ^ , of equation (25.43) is repre- sented by a sinusoid, while the right-hand aide is repre¬ sented by two straight lines of slopes ± . If a-\/2mUa the absolute value of the slopes of the angle of incli- xa nation of these lines is less than 2/7t, they have one or several common points with Fig. 39 the sinusoid in the quadrants corresponding to the roots of (25.41). The trivial point of intersection xa =0 does not count because, for x == 0, the wave function is zero everywhere. Thus, in a weU of finite depth of the form considered, there are only several energy eigenvalues. o'\/2mt7j ^ Tt ’ 8 ma<‘ ’ there are in general no points of intersection of the straight lines with the sinusoid corresponding to energy eigenvalues. In Fig. 39 the points of intersection in the even quadrants are marked by small circles. Sec. 25] CERTAIN PROBLEMS OP QUANTUM MECHANICS 263 Finite and infinite motion. We shall now relate the shape of the energy spectrum to the t 3 ^e of motion. For ^ >Uq the solution outside the well is of the form (25.33). It remains finite also for an infinitely large x. Therefore, the integral J | ij; | * da; taken over the region of the well is infinitesimal compared with the same integral taken over all space. In other words, there is nothing to prevent the particle going to infinity. Such motion was termed infinite in Sec. 6. For Uq, the solution (25.38), if it exists, is exponentially damped at infinity. Hence, the probability of the particle receding an infinite distance from the origin is equal to zero—^the particle remains at a finite distance from the well all the time. This motion was termed finite in Sec. 5. Thus, infinite motion has a continuous energy spectrum while finite motion has a discrete spectrum consisting of separate values. If the depth of the well is very small, the finite motion may bo absent. It has no comiterpart in classical mechanics. Finite motion is always possible in a potential well if | ^ | < 17. The result that we have obtained does not only refer to a rectangular potential well. Indeed, if the potential energy is taken to be zero at infinity, then the solution with positive total energy is of the form (26.33) for sufficiently large x, while the solution with negative total energy is of the form (25.38). The latter contains only one arbitrary constant while (25.33) contains two constants. Both solutions must be extended to the coordinate origin in order that the condition <|> (0) = 0 can be satisfied at the origin (we consider that x is always greater than zero). Obviously, if we have two constants at our disposal we can always choose them so that the condition t{j (0) = 0 is satisfied.* Contrarily, a solution of the form (25.38) containing one constant becomes zero at the origin only for certain special values of x. A continuous spectrum corresponding to infinite motion may be accounted for in the following way. A free particle moving in unbound¬ ed space has a continuous spectrum. The wave function of the particle in infinite motion differs from the wave function of a free particle only in the region of a potential well. But the probability of finding the particle in this region is infinitesimal if the whole region of motion is sufficiently large. Therefore, the wave function for infinite motion coincides with the wave function of a free particle in “almost” the entire space, i.e., in that region of space for which the probability of finding the particle is equal to unity, and the energy spectrum turns out to be the same as for a free particle. The wave function in a region where the potential energy is greater than the total energy. If Uq tends to infinity the function outside the * If ^ (0) = (0) -H (0), then . 264 QUANTUM MECHANICS [Part III well very rapidly tends to zero. In the limit Uq ->■ oo, it tends to zero however close to the boundary x—a, thereby giving the boundary condition (26.1). In the case of a finite C/q ^^e wave function outside the well does not become zero at once. Therefore a finite probability exists that the particle will be outside the well at a finite distance from it. This would have been completely impossible in classical mechanics, as is obtained from (25.38) in the limiting transition h -»0 for k = oo and (j; however small outside the well. This, naturally, should be the case: if the particle is situated outside the well its kinetic energy is S — f/fl < 0. But the velocity of such a particle is an imaginary quantity. In classical mechanics it means that a given point of space is absolutely unattainable for the particle at the given value of its energy S. In quantum mechanics, a coordinate and velocity never exist in the same states as precise quantities. Earlier, we interpreted this in terms of the uncertainty relation, i. e., we considered cases for wliich precision in the concept of velocity for a certain state was restricted by the limits . However, this is a lower limit and has to do with particles which are almost unaffected by forces. The appear¬ ance of an imaginary velocity in the equation for a boimd particle shows that the very concept of velocity is not applicable to a region of space, however large, for which U >S. We can express this differ¬ ently by saying that, for V >S, the uncertainty in the kinetic energy is always greater than the difference U — To summarize, in classical mechanics there is no counterpart to the motion of a bound particle outside a well. Exercises I) The potential energy is equal to zero for x<0 and equal to for x>0 (the potential threshold). Incident from the left are particles with energies S’ > U„. Find the reflection coefficient. The wave function on the left is j 2^ _ . px On the right, above the threshold, the function is ■ p'x Cj e ** , where p' — ■\/1m (S — U). Find the ratio |0*|®/|CiP from the boundary conditions at x—0, i.e., the ratio of the squares of the amplitudes of the reflected and incident waves. The ratio is equal to unity for S< Uo. 2) The potential energy is equal to zero for x<0 and for x>a. U — Uq for 0“‘ ’ ® (x + ik)^e -xa — (x — ik)^e^‘> The particle flux on the left and right of the barrier is, respectively. • hk „ m |CP), j: hk m Substituting G and G 3 it is easy to see that both the expressions for flux coincide as expected. If xai> 1, i.e., the barrier is transparent to a very small extent, wo hav’o G -1 , Ca = - — - 6-*"6-xa . X Thus the flux diminishes exponentially with the thickness of the barrier. It will also be noted that the total particle flux through the barrier is pro¬ portional to the particle density in front of the barrier, because the boundary conditions are linear and homogeneous with respect to the wave functions. By specifying the amplitude of the wave fimction on the left we determined the density and flux of the particles. 3) Verify the orthogonality property for wave functions (the exercise in the preceding section) for a particle in a box of finite and infinite depth. Sec. 26. Harmonic Oscillatory Motion in Quantum Mechanics (Linear Harmonic Oscillator) The wave equation lor an oscillator. In Sec. 7 we considered har¬ monic oscillations with one degree of freedom. The Hamiltonian func¬ tion of this system, called a linear harmonic oscillator, is of the form J^ = P'‘ 2 m 2 ( 26 . 1 ) Forming Hamilton’s equations, we obtain p = - d3^ dx = — • _ _ p dp m ’ Eliminating p, we arrive at the usual equation of harmonic oscillations (7.13): x + = 0. 266 QUANTUM MBOHAiaCS [Part III In quantum mechanics the wave equation corresponding to this motion has the form [see (24.22)] . TO 6 >“a:“ , 2m dx^ ' 2 ^ (26.2) Indeed, since the motion has only one degree of freedom, instead of the Laplacian A, we must simply write the second derivative. The potential energy is equal to —^— • Let us now introduce other units of measurement, in particular, we shall take the unit of length equal to 1/—^, so that x = (26.3) The quantity ^ is dimensionless. The derivative dtjl dx is equal to Further, we put d^ _ / W6> d^ dx h * 2S = £ • A. The transition to another dependent variable. It appears convenient to introduce a new dependent variable: Whence ^_ If :? 2 e‘ il 2 sr(?)- -^1 ^ S’i?) + e e 2 g(l) dl ^ dl + d£“ • (26.7) (26.8) We substitute (26.8) in (26.6) and perform the necessary rearrange¬ ment. The new dependent variable g (^) having been introduced, the equation assumes the form Sec. 26] HABMOmO OSClIiliATOBY MOTION 267 + = (26.9) Integration in the lorm of a series. It is possible to integrate equation (26.9) by expanding it in a power series of the form; oo ? (^) = ffo + ?! ^ + ?2 + 9^3 + • • • (26.10) n-0 In order to determine the coefficients of the expansion gr«, we must substitute the series (26.10) into equation (26.9), differentiate it by terms and compare the expressions for the same powers of 5- The first derivative is OO = 9-1 + 2sr,5 + 3S-3 5* + ... n-1 SO that oo + ... =2j2ng„l''. (26.11) n ™ 1 The second derivative is oo = 2sr3 + 6g^l + • (26.12) )c = 2 In the last summation we changed the summation index, denoting it by the letter k. We shall now revert to n, assuming k — 2=n, k — n+2. Then oo =2’(” + 2) (« + + (26.13) «=o Now substituting (26.13) and (26.11) into equation (26.9) and collecting coefficients of we obtain oo (w + 2) (n + l)g „+2 + 2ng„ — {e — 1)!7„] = 0. (26.14) M = 0 We know that for a power series to be equal to zero, aU its coefficients must convert to zero. Thus, 9 n + 2—S-n 2) (n + ry* (26.16) In this way the expansion proceeds in powers of (^^) because the coefficients gn go alternately. Examining the series. Let us assume initially that go¥=0. Then, from equation (26.13), we find in turn g^, g^, .. .,gzk. Not a single odd 268 QUANTXJM MECHANICS [Part III coefficient will appear in the series if pi=0. On the contrary, if gQ = (i, gi^O, then no even coefficients will appear in the series; for this reason it is sufficient to examine solutions containing only even or only odd powers of To be specific let us first take the series in even powers. Let us examine the behaviour of the series (26.10) for large values of Terms involving high powers of i. e., large n are then predomi¬ nant. But if w is a large number then, in equation (26.16), we can neg¬ lect all constant numbers where they appear in the sum or, difference with n. If n is large the equation will take on the form 2 gn + i — ' (26.16) Let n — 2n' so that n' now changes by unity only. Putting this in (26.16), we obtain gn'-v\ = -Wn' (26.17) where we have introduced the notation g'n' = g+i: gl, 1 — n' (n’ — l) in' —2) ... 1 (26.18) Whence the expression for gr(^) in the case of large ^ is of the form: oo oo n'»» 0 0 Thus, the asymptotic expression for g(^) is the exponential function But then, in accordance with the definition of g(^) (26.7), the asymptotic form of ->00 —>00 (see exercises of Sec. 39). Sec. 27] QUANTIZATION OF THE EEECTBOMAGNETIC FIELD 271 See. 27. Quantization of the Electromagnetic Field The electromagnetic field as a mechanical system. An electromagnetic field in a vacuum may be regarded as a mechanical system; this was shown in Sec. 13. It possesses a Lagrangian function, action, and so on. We are, therefore, justified in posing the problem of quantization of this system, i.e., applying quantum mechanics to it. The basic difference between electrodynamics and the mechanies of point masses is that the degrees of freedom of an electromagnetic field are distributed continuously: in order to specify the field at a given instant of time, we must define its value at every point of space. In this sense electrodynamics resembles the mechanics of liquids or elastic bodies, if one regards them as continuous media ignoring the atomic structure of the substance. The degrees of freedom of a field are labelled by the coordinates of points in space, while the amplitude values of the potential are generalized coordinates [see (13.2)]. Poten¬ tials are usually chosen as generalized coordinates because they satisfy second-order equations in time, as do generalized coordinates in mechanics. The potentials satisfy the Lorentz condition, which reduces to div A = 0, provided the gauge transformations are chosen so as to eliminate the scalar potential. The electromagnetic field coordinates defined in this way are not independent of one another. Indeed, the equations of eleetrodynamics involve coordinate derivatives, i.e., differences of field values at infinitely close points. In this sense, field equations resemble the equations for coupled oscillations: they are linear, but each one in¬ volves several generalized coordinates instead of one. The equations for coupled oscillations can be reduced to normal coordinates which are mutually independent. The same can be done with the wave equations of electrodynamics, thus separating the dependent variables therein. This considerably simplifies the application of quantum mechanics to radiation. Clearly shown here is the generality of the methods of analytical mechanics: they permit determining generalized coordinates and momenta in such manner that quantum laws can then be applied uniquely. The electromagnetic field in a closed volume. We must first of all represent the electromagnetic field as some kind of closed system, since quantum mechanics is most conveniently applied to such systems. We can assume, for example, that the radiation is contained in a box with mirror-type reflecting walls. At the walls of such an imaginary box (a: = 0 or x—a^, y — 0 or y—a^, 2=0 or 2 =a 3 ) the normal components of the Poynting vector U become zero. However, it is simpler to suppose that the field is periodic in space, and the lengths of the periods in three perpendicular directions are equal to the di- 272 QUANTUM MECHANICS [Part III mensions of the box; the period of the field along x is equal to a^, that along y is equal to a^, and that along z is equal to a^. In other words, A (x, y,z) = X{x-\- Ui, y,z) = A (x, y + a^, z) = = A{x,y,z + a^). (27.1) We have, as it were, divided space into physically identical regions, after which it is sufficient to consider a single region. The solution of equations describing a harmonic field in free space was found in Sec. 17 [see (17.21)]. Introducing a time dependence into the amplitude factor, we represent the potential in the following form: A (k, r, t) = A^ (<)e*'‘' + A* (<) , (27.2) where its reality is shown explicitly. The potential satisfies the Lorentz condition which, for a plane wave, can be reduced to the form div A = 0 (since 9 = 0); whence, by (11.27), we obtain divA(k,r,<) = div(A^e''‘') + div(A*e-*'“) = = (A,^Ve'''“) + (A* Ve-'>“) =i(kAJe‘'“—i(kA’)e-““-=0. In order that this equation be satisfied for all r, the coefficient of each exponential term must convert to zero. In other words, the vectors Ak and At are perpendicular to the wave vector k: (kAJ=0, (kA*) = 0. (27.3) For each k there exist two mutually perpendicular vectors AS (u = 1,2 ) corresponding to two possible wave polarizations. Any vector in a plane perpendicular to k can be resolved into Ak^ and Ak^*. We shall now apply the periodicity condition (27.1) to each term of (27.2) separately. For the first term we obtain gi(jtxx + ki/r + ktz) — gi[“). (27.8) k.a The magnetic field is determined from (21.28) and (12.28): H = rot A =27([Ve'‘‘'. A"] + [V e—A-*]) = k,o = i27([liA3 [kAg*]e—■'“■). (27.9) k,cr Let us now calculate the field energy. According to (13.21) (27.10) To obtain we perform summation over k, k', er, and o': E^=-Z- k, k'o, o' (A? A°;e' (k +k') r . • A“A®'*cOk- - k') r . — Ag'Aj^e-'O'- + A2’A?;*e-'(‘‘ t “'J'’). (27.11) It is expedient, when integrating E^ over a volume, to change the order of summation and integration, each volume integral being 18 - 0060 274 QUANTUM MECHANICS [Part III reduced to the product of three integrals of the following type: .1 27i:» (/ii + Hi) (27.12) 0 « for Wj + # 0. If Wj + wj = 0 this integral is equal to a^. Therefore the triple integral assumes one of two values: ( 27 . 13 , It follows that in the expression for the double summation with respect to k and k' becomes a single one after integration, and wo must replace k' by —k in the terms involving the product At AS'. In terms containing Af Af'* we replace k' by k due to the factor g—ik'r "Phus. - A°A°'’ k k k, 0. The relation (27.24) has a counterpart in classical electrodynamics; the energy density of an electromagnetic wave was shown in Sec. 17 to be related to its momentum hy means of the factor c. In the limitmg transition to classical theory, the energy of each quantum is regarded as infinitely small while their number iV^t, o is infinitely large, so that the wave amplitude remains finite. Occupation numbers. Passing from Cartesian coordinates to new independent variables (the components of the wave vector k), wo renumbered the radiation degrees of freedom, the quantities Qk now being the generalized coordinates. The state of a field is specified if all the “occupation” numbers Nk, a are known, because the number Nk, a defines the quantum state the given harmonic oscillator is in, i.e., the number of quanta in the state k, a. The numbers Nk,a may be regarded as the quantum variables of an electromagnetic field. When a field interacts with a radiator (for example, an atom) these numbers change. For example, if the number Nk,o has increased by unity this means that a quantum of corresponding frequency, direction, and polarization has been emitted. The ground state of an electromagnetic field. Let us now examine expression (27.23) when Nk,a — 0. In other words, let us determme the ground state energy of the electromagnetic field. According to (27.23) it is ^(0) =-2-27-.3- (27.26) «i, "3. CT But since the numbers Wj, n^, and run through an infinite set of values, the sum (27.25) is infinitely large. It must be said that in this case the theory is not fundamentally defective because the zero energy itself (27.25), does not appear in any expression; the field energy is always measured from the ground state. At the same time, the ground state of the quantum oscillators of an electromagnetic field leads to actually observable effects because the amplitude of a harmonic oscillator in the ground state is not equal to zero. It takes on all possible values, and, in accordance with (26.22), the probability of a certain value of Q is proportional to | (Q) j®. In an electromagnetic field, the part of the oscillator coordinates is played by the generalized coordinates Q, in terms of which the field amplitudes are expressed linearly. Therefore, one can by no means assert that the field amplitudes are equal to zero in the ground state of an electromagnetic field (i.e., in the absence of quanta). The prob¬ abilities of definite amplitude values are given by the harmonic- oscillator wave functions. These functions are equal to e in coordinates. 278 QUANTUM MECHANICS [Part III Electron-level shift produced by the ground state of a field. The ground state of an electromagnetic field affects observable quantities. One of the most important effects of this tyjie consists in the following. IjCt an electron move in the iiotential field of a nucleus. The value of tlic electromagnetic potentials of a field acting on the electron is usually chosen as A = 0,

U, when the root is extracted from a positive quantity, but also when S 0, the wave function will be damped in this region infinitely quickly—like e ~°°; and this denotes the unattainabUity of points where for classical motion S' S extended rightward to infinity. Therefore, the wave function became zero at infinity. Considerable interest is attached to another problem, in which the potential energy at a certain distance away from the well again becomes less than the total energy. This is shown in Fig. 42. U >S for the region < a; < a:^. Therefore, in classical mechanics a particle situated to the left of x — Xi cannot under any circum¬ stances attain the region x >X 2 , from which it is separated by a potential barrier. In Fig. 42 quantum mechanics, the wave function does not become zero between Xi and x^, since this region is finite (see exercise 2, Sec. 26). In the approximation (28.1), the exponent in the equation is a real quantity when x < ar^. Therefore, the modulus of t|; remains equal to unity: 282 QUANTUM MECHANICS [Part III S 5 On the other hand, between x-y and x^, the modulus of t{) decreases according to the law \ m (U — dx Xl j = e AJv2 m(U- dx xi (28.5) At x = X 2 , in comparison with the point a: = a;j it diminishes in the ratio X, - I \/2m(U^^ dx B=e i (28.6) after which it again stops changing, since 8 becomes a real quantity. Hence, the square of the modulus of the wave function diminishes between Xy and x^ according to the expression determined by the quantity B. A more precise theory provides a correction factor for B, though fundamentally, the function is determined by B alone. B is called the barrier factor. Somewhat later, it will be explained how the B factor is related to the penetration probability through the barrier in unit time. The quasi-classical approximation is feasible only when the order of magnitude for the action is large compared with h. This was men¬ tioned in Sec. 24, when the conditions for the limiting transition to classical mechanics were being determined. For this reason, equation (28.6) may be used only when the exponent is large compared with unity. If it is comparable with unity, the penetration probability through the barrier must be evaluated by means of precise wave func¬ tions. The Mandelshtam analogy. In wave optics there is an analogy to the passage of a particle through a potential barrier. When a lighi wave falls on the boundary of a medium of small refractive index fronr a medium of larger refractive index at an angle whose sine is greater than the ratio of the indices of refraction, there occurs total internal reflection in accordance with the laws of geometrical optics. If the problem is solved in strict accord with wave optics, on the basis of Maxwell’s equations (see exercise 2, Sec. 17), then it turns out that the wave penetrates somewhat into the second medium, but dies out exponentially in it. L. I. Mandelshtam took notice of this analogy between quantum mechanics and wave optics. It can be applied in the following manner. Let us imagine two optically dense media separated by a layer optically less dense. Let a light ray fall on the interface of the media at an angle to the normal larger than the angle of total internal re¬ flection. According to geometrical optics, the ray should be complete- Sec. 28] QUASI-CI.AS8ICAL APPROXIMATION 283 ly reflected by the layer, and it is absolutely immaterial whether or not there is a denser medium beyond the layer, or whether the re¬ flection occurs from an infinitely thick, nondense medium. Similarly, a particle in classical mechanics is completely incapable of penetrating the barrier. According to the laws of wave optics, light penetrates into a nondense medium, but dies out in a thickness comparable with the wavelength., Therefore, if the second dense medium is situated closely enough, part of the light “seeps” into it. The classical expression for the amplitude of a light wave may be regarded as the wave function in relation to a light quantum. The transition from quantum theory to classical electrodynamics consists in considering the occupation numbers as large (see Sec. 28). Then the corresponding field amplitudes change to classical ones. For this reason the Mandelshtam analogy represents an example of quanta penentrating through a barrier. As has already been pointed out, the limiting transition for electrons occurs differently; it corresponds to the transition from wave optics to geometrical optics. Therefore, in the classical limit, electrons do not penetrate the barrier. The existence of penetrations through a barrier clearly indicates that the concept of a trajectory is sometimes completely inapplicable in the case of quantum motion. A trajectory extended under the barrier would lead to imaginary velocity values. Alpha disintegration. Passage through a potential barrier enables us to explain one of the most important facts of nuclear physics, that of alpha disintegration. The nuclear masses of heavy elements with atomic numbers greater than that of lead satisfy an inequality of the form ( 21 . 12 ): TO (A,Z) >m (A — 4, Z —2)-f-m (4, 2). (28.7) Here A is the atomic weight and Z is the nuclear charge (i.e., the atomic number in the Mendeleyev table). Thus, to (4,2 ) is the mass of a helium necleus with atomic weight 4 and atomic number 2. Such a nucleus emitted during alpha disintegration is called an alpha particle. All that can be seen from equations (21.12) and (28.7) is that the spontaneous decay of a nucleus of mass m {A, Z) is possible, though no indication is obtained about the time law of disintegration. The nuclei of certain elements have mean decay times of lO^® years while others have decay times of about 10~® sec, which is a difference of 23 orders of magnitude. It will be noted that the energy of the alpha particles emitted differs here by a factor of only two. From experiment it turns out that the logarithm of the mean decay time of a nucleus is inversely proportional to the alpha-particle velocity. It is this logarithmic law that corresponds to the difference of 23 magnitudes. It is accounted for by the difference of barrier factors which depend exponentially upon the energy. 284 QUANTUM MECHANICS [Part III The potential-ener^ curve. At large distances from a nucleus, an alpha particle experiences a repulsive force of potential energy U = 2 (. 2 - 2 ) r (28.8) [cf. (3.4)]. At small distances, attractive forces must act because, otherwise, the nucleus {A, Z) could not exist at all. We do not know the force law (i. e., the shape of the potential-energy cmve when the alpha particle is situated sufficiently close to the nucleus) and, therefore, in Fig. 43 we draw it at will. In this we must be guided by the following considerations. The special nuclear forces which hold the alpha particle in the nucleus before emission have a small radius of action, so the potential-energy curve has the form of a “potential well.” Motion inside the nucleus corresponds to motion inside such a well. The transition region from the well to the Coulomb curve is not very essential for final results, i. e., it little affects the exponent of the barrier factor. The barrier factor for alpha disintegration. The energy of an alpha particle is positive at an infinite distance from the nucleus. It is this that signifies that the nucleus is capable of alpha disintegration, i.e., the alpha particle can move infinitely. In order to find the prob¬ ability for alpha disintegration, we must calculate the barrier factor B in accordance with (28.6). Because nuclear forces are short-range forces, the transition region is small and we can extrapolate the Coulomb law, without sensible error in the integral, up to the point r = where S becomes greater than U. Point is the effective nuclear radius determined from alpha disintegration. Other data concerning the nucleus lead to somewhat different values for the respectively determined effective radius. This is understandable since is obtained on the particular assumption that the Coulomb law is valid up to the region for which the potential energy curve is taken in the form of a well with steep sides. And so we determine the barrier factor according to the equation -^-7 ~ • Substituting, in place of Aw, the equivalent th A (in this case) quantity ■ , we arrive at the following estimate: A2nh. (28.15) Tliis is the uncertainty relation for energy. The measure of uncertainty for the energy is the quantity P, i.e., the inverse value for the probability of decay in unit time. (28.15) should he formulated thus; an energy of a state existing during a limited time interval A< ^ It h is determined within the accuracy of the order of . Only the energy of a state which exists an unlimitedly long time is fully deter¬ minate. The meaning of the uncertainty relation for energy. The meaning of the uncertainty relation for the coordinate and the momentum (23.4) is not analogous to the meaning of (28.15). The estimate (23.4) expresses the fact that the coordinate and momentum do not exist in the same state; (28.15) signifies that if a state of the system has a finite duration At, then its energy at each instant of time within the interval At is not determined exactly, but is only contauied Avithin a region of the order of P. The quantity P is termed the level width of the system. The concept of level width can be applied to any states of finite duration and not only to alpha disintegration. For example, the energy level of an atom in an excited state has a definite width, since an excited atom is capable of the spontaneous emission of a quantum. Explanation of the level width. We shall now show how the level width of a nucleus capable of alpha disintegration can be found by considering the wave function variation under a potential barrier. In was shoAvn in Sec. 25 that infinite motion has a continuous spectrum. The motion of a system with a potential barrier is infinite because an alpha particle is capable of going to infinity. It follows, strictly speaking, that a nucleus capable of alpha disintegration shoidd have a continuous spectrum. * A wave with definite frequency is termed “monochromatic.” A mono¬ chromatic wave corresponds to a single colour (chromos is the Greek for colour). Sec. 28] QTTASI-CL,ASSICAIi APPROXIMATION 289 Let us now evaluate the energy level widths F for alpha-radioactive nuclei. From (28.16) it follows that even for a nucleus with a very short alpha-decay time (i~10-®sec) F^IO-®* erg~0.6 x 10“^®ev. How is it possible to combine a continuous spectrum with such a narrow energy interval ? The solution to the wave equation between and r.^ is of the following form: —-L j V 2m(u—g) dr T I 2m*_ — \/ 2^/m 01* From this = ha> which is fortunately correct for all n and not only for n > 1. 2) Find the approximation that follows (28.3). We look for IS in the form S = S# + hSi- Then = e h h + 'H-ySo'4' 2m - U)<^. The zero approximation gives from (28.3). The first approximation yields = or S, = iln . s„ 1 I - so that iji = —e " . V *’o 3) Find the factor B for a barrier of the form 17 = 0 when x<0, C7 = Ug —oa: when a: > 0, ^ Sec. 29. Operators in Quantum Mechanics Momentum eigenvalues. In a number of cases we are able to deter¬ mine energy eigenvalues from the wave equation (24.22). However, it is very important to find the eigenvalues of other quantities too: linear momentum, angular momentum, etc. To do this, it is con¬ venient to proceed from the form of in the limiting transition to classical mechanics; 292 QUANTUM MECHANICS [Part III h d Let US apply the operation to both sides of equation (29.1), i.e., we take the partial derivative with respect to x and multiply by : A PA. = AA e' ^. i dx dx (29.2) But in the classical limit S becomes the action of the particle, while becomes the component of momentum p* [see (22.9)]. Therefore, the equation for the momentum eigenvalues that yields the correct transition to classical mechanics is of the following form: h 5']; i dx (29.3) where p* is the eigenvalue for the a;th momentum projection. Momentum and energy operators. Let us compare equation (29.3) with the wave equation (24.22): +(t-^) +(t:£') J <1^ + = <^<1^ • (29.4) Here, the symbol | <}' denotes and similarly for 4^ > (aj) ')'• In order to find the energy and momentum eigenvalues we must perform a definite set of differential operations and multiplications by the function of coordinates in the left part of the equation. But these sets are connected in a very curious manner, as will now be the momentum h a shown. We shall call the symbol , multiplied by -4 operator applied to a wave fimction. Instead of we will symbolically write p*. Then, it will be necessary to rewrite equation (29.3) as p»4 = • (29.5) This equation denotes exactly the same as (29.3), though the sym¬ bolic notation p^ should emphasize that the corresponding operation is applied in order to find the momentum eigenvalues. The operation on the left-hand side of (29.4) we shall also symbol¬ ically call We write and not because the energy is assumed to be expressed in terms of momentum, similar to the Hamiltonian func¬ tion if. Then, in shorter notation, (29.4) appears as 4 . (29.6) is called the Hamiltonian operator, or the energy operator. Sec. 29] OPBB.ATOBS IN QUANTUM MECHANICS 293 Comparing (29.4) and (29.3), we see that the momentum and energy operators are related by the same equations as the corresponding quantities: + + U ■ ( 29 - 7 ) We have written U instead of simply U in order to emphasize that in this equation the expression U is not regarded as an independent quantity but, instead, as an operator acting upon <{/, i.o., a multipli¬ cation operator of t{; by U. Equation (29.7) is symbolic. It is under¬ stood that both sides are applied to ({/. The meaning of operator symbolism. The usefulness of an abbreviat¬ ed operator notation in quantum mechanics consists in the fact that the equations thus become more expressive. The relation between quantum laws of motion and classical laws, which are limitmg cases with respect to the quantum ones, can be best of all seen in operator notation. If in classical equations relating meclianieal quantities we replace the momenta by their operators, we then obtain correct operator rela¬ tionships of quantum mechanics. The limiting transition to classical mechanics restores the usual relationships between quantities. Indeed, in the limitmg transition (29.1), the operator p=-^ V If we must perform the limiting transition for p®, then we need to differentiate only the exponential each time, because this yields the quantum of action in the denominator. For h-^0, only terms with the highest degree of Ji in the denominator remain, and it is these very terms which are obtained in replacing the operator p by the quantity V8 (i.e., by the classical momentum vector). We had an example of such a transition in Sec. 24 [see equations (24.13) and (24.14)]. The angular-momentum operator. It is now easy to define the angu¬ lar-momentum operator. We shall begin with one component Mz. It is clear from Sec. 6 that the angular momentum Mz is at the same time a generalized momentum corresponding to the angle of rotation about the 2 -axis, i.e., Mz = p^. Then, from (10.23) ' (29.8) Therefore, in quantum mechanics the operator p, must be of the form K-I-A. ,29.9) At the same time, m accordance wdth classical mechanics, the pro¬ jection Mz is related to the momentum projections thus: Mz = xpy — ypx. (29.10) 294 QUANTUM MECHANICS [Part III It follows that there must exist an operator relationship p, = J/z = 4 ~ ^ = a:py - . (29.11) Let us check to see that the definitions (29.9) and (29.11) do, indeed, coincide. Let us pass to cylindrical coordinates: From this we have X — r cos 9 , (29.12) y — r sin 9 . (29.13) Sij) 0 i}i dr . Si]) Stp dx dr dx S 9 dx ’ (29.14) 3 i]i Sij) dr 9ij' dy dr dy ' d2itik probability ampli- (29.19) (29.20) where k is an integer of any sign or zero. Whence we obtain the eigen¬ values of Mz'. Mz-=hk. (29.21) Mz is called the orbital angular-momentum projection for a particle. We shall see in Sec. 32 that a particle can have an angular momentum connected with its internal motion, which is not described by the wave function (29.18). Here, we have proven that the orbital angular- momentum components can only assume values which are whole multiples of h. The Stern-Gerlach experiment. The discreteness of the angular- momentum spectrum is confirmed by direct experiment. The idea of the experiment consists in the following: a direct relationship exists between the orbital angular-momentum projection and the magnetic moment projection [see (15.25)]: A narrow beam of vapour of the substance under investigation is passed between the poles of an electromagnet in a strongly inhomo¬ geneous field; to achieve this, one of the poles may be made tapered. The particles—^in the Stem-Gerlach experiment, they are atoms— enter the field parallel to the edge of the taper, i.e., they move in a direction perpendicular to the plane of the lines of force of the field. The plane of symmetry of the field passes through the edge of the taper and the initial direction of motion of the particle. We assume the a-axis to be perpendicular to the edge of the taper and to lie in the plane of symmetry of the field. If the mechanical moment of the electrons in the atoms has only discrete, integral projections on the 2 -axis, then the magnetic moment of the atoms is established in several definite ways. The deflecting force acting on the magnetic moment in a magnetic field is, by (15.40) dH , eh 8H 296 QUANTUM MECHANICS [Pait III In the plane of symmetry of the field, H is directed along the z-direction and depends only upon z. Since the angular momentum can only have a definite set of values, the deflecting force acting upon the atoms in the beam is also not arbitrary but has a very definite value for particles with respective angular-momentum projection M^^hk. It can be seen from (29.23) that the force is a quantity which is a multiple of • Therefore, the particles in the beam experience only those deflections in the magnetic field which correspond to the possible values of the force (29.23). In other words, the beam is split into several separate beams and does not proceed continuously, as would be the case for any nonintegral projections M^. Where the beam is formed, each particle was given a certain angular momentum. Motion in the magnetic field makes it possible to measure the projection of this angular momentum Mz in the direction of the field. The impossibility of the simultaneous existence of two angular- momentum projections. From the fact that the angular-momentum projection on any axis is integral, it follows that the angular mo¬ mentum does not have, simultaneously, projections on two axes in space. Indeed, in the Stern-Gerlach experiment, the z-axis is absolutely arbitrary. We could have measured the angular-momentum projection on some axis in space and then pass the same beams through a magnetie field making a very small angle with the field in which the first measurement was performed. Both measurements wUl give only integral projections of the angular momentum. Both one and the same vector cannot simultaneously have integral projections on infinitely close, but otherwise arbitrary, directions; when the first measurement was performed, the angular momentum had a projection only on the first direction of the field, and, correspondingly, in the second measurement, it had projections only on the second direction of the field. Similar to the way that coordinate and momentum do not exist simultaneously, it turns out that two angular-momentum projections do not exist in the same state. The simultaneous existence of two physical quantities. We shall consider from a general point of view the question of which quantities of quantum mechanics can exist in the same state of a system. Let us suppose that in a certain state described by the wave function iL there simultaneously exist two physical quantities X and v. This means that the wave function t|; is an eigenfunction of the two operators X and V. It satisfies two equations XiJ/ = XtJ; V . and (29.24) (29.25) Soc. 29] OPERATORS IN QUANTUM MECHANICS 297 X and V are, speaking generally, differential operators; X and v are numbers. Let us apply the operator v to (29.24). Since there is a number X on the right, it can be put on the left of the operator sign v : vXtj; = vX4< = Xviji = Xvij;. (29.26a) In the last equation we made use of (29.26). We shall now apply the operator X to (29.25): Xvt}; = Xv(j> = vX(J; = vX^*. (29.26b) Let us subtract (29.26b) from (29.26a): 5X -pr- 1 . 1 \ Od O 9 / Applying M* + t'My to M*—iMy, wo must observe the order of the “factors” S " ' d -—and ci'P, cot * and-;--. We obtain 09 oft (Mx+iMy)(Mx-iMy)^ = -A2 .0 a .. a.a . „a.a\ i- 5 ^cot ft-;-h te'vcot ft-;^e->9-5-;^ -b e'9cot^ft = Finally, / a* left* 'M 09 ’'"09“ 'Oft ' '09 )• M*. / 1 d ■ „ 3 . 1 a« ' \smft aft 47 * fa. '*'!" • tk ^ aft sm®ft a 9 ^ A a This expression is obviously commutative with Mz = —. This was shown in the present section in another way. Sec. 30. Expansions into Wave Functions The superposition principle. One of the most fundamental ideas of quantum mechanics consists in the fact that its equations are linear with respect to the wave function This result proceeds from the whole set of facts that confirm the correctness of quantum mechanics, in the same way as an analogous result in classical electrodynamics (see Sec. 21), which is also a generalization of experience. For example, the diffraction of electrons shows that the amplitudes of wave functions are combined in the same simple way as the ampli¬ tudes of waves in optics; diffraction maxima and minima are situated at the same positions, which are determined only by the phase relation- 302 QUANTUM MECHANICS [Part III ships, indeiJendently of the wave intensities. All this points to the linearity of wave equations; the solutions of nonlinear equations behave in an entirely different manner. The sum of two solutions of a linear equation again satisfies the same equation. It follows from this that any solution of a wave equation can be represented in the form of a certain set of standard solutions, similar to the way that, in Sec. 18, a travelling nonperiodic wave was represented by a set of travelling harmonic waves (18.1). The statement concerning the possibility of representing a single wave function in terms of the sum of other wave functions is called the superposition principle. The Hermitian property ol operators. Wave fimctions are usually represented with the aid of the sum of eigenfunctions of certain quantum-mechanical operators. In the present section it will be shown how such expansions are performed. First of all, however, it is necessary to establish certain general properties of the operators whose eigenvalues are physical quantities. Obviou.sly, these eigenvalues must be real numbers, although the operators themselves may depend explicitly upon i=V — 1 [see (29.3), (29.10)]. We shall consider the equations for the eigenfunctions of the operator X and another equation involving its conjugate: Xt|; = X(]^, (30.1a) X*tj;* = X*(j;». (30.1b) We must find the condition for which the eigenvalues of the operator are real numbers, X*=X. To do this, we multiply (30.1a) by (j;* and (30.1b) by integrate over the whole range of the variables x (upon which the operator X depends), and subtract one from the other. Then we obtain J((|;* Xt}'— (j^X*(};*) dx — (X — X*)j4'*^^da:. But the integral of cannot be equal to zero, since is an essentially positive quantity. The eigenvalue X is, by definition, real, i.e., X=X*; therefore we arrive at the relation J(({'*X4' — ~ ® • (30.2) Equation (30.2) can be regarded as a condition imposed upon the operator In fact, however, we must demand that the operator X satisfy the equation (30.2) not only for its own eigenfunctions (X, x) and (j; (X, x), but also for any pair of functions x* (*) ^ (*). Pro¬ vided these functions satisfy the same conditions of being finite, continuous, and single-valued as the eigenfunctions (|> (X, x): Sec. 30] EXPANSIONS INTO WAVE FUNCTIONS 303 —= 0 . (30.3) The necessity of such a condition will be explained later in this section. An operator for which equality (30.3) is satisfied is termed Hermitian. In equation (30.3), dx is an abbreviated notation for dV=dx dy dz, if the integration is performed over a volume (X-^.^), orrf(p,ifX = Mz, etc. The Hermitian nature of the operators p*, Mz ,... is easily verified by integrating by parts. For example, 0 0 0 0 The eigenfunctions of the operator Mz must satisfy the requirement of uniqueness (29.19); hence (0) =;^* (27t), tj; (0) (2:1:) similar to the eigenfunctions of the operator Mz- Therefore, the integrated quantity becomes zero. The operator — is Ml, so that 27c 2n j'x* Mz^ d

(30.12) The meaning pf the expansion coefficients. We have seen that a state tj' (^) is represented as a superposition of states with definite values of the quantity X. The component of the wave function which corresponds to this value of X is cx^l^ (X, z ). It represents the probability amplitude for the given value of the quantity X in the state (x). In order to find the probability itself, Wx, of the occurrence of quantity X, we must ehminate the coordinate dependence, since X and x do not exist in the same state. To do this, let us integrate the probabihty density of the state with a given X, i.e., | Cx | <^ (X, x)^, over all x. From the normalization condition for eigenfunctions we obtain X)\^dx= |cx I*. (30.13) The quantities Wx= 1 cx 1^ have a basic property of probability: their sum is equal to unity, provided the function itself satisfies the condition of normalization (24.18). Indeed, 1 = J| (Kx )\^dx =J I 1®^* = X =I®/1(^. *) H 'f'* 'f' • X X X'^SX But on the orthogonality condition (30.6) a double summation is equal to zero. From this, in accordance with (24.18), it follows that 2'|cx|*=i;«^x = 1. (30.14) Thus, the coefficient Cx should be regarded as the probability ampli¬ tude, similar to (j; (x). But | (a:) is the probability of detecting a particle with coordinate x independently of X, while |cxl® is the prob¬ ability of finding it with a given value of the quantity X independently of X. Expansion in angular-momentum projection eigenfunctions. The atomic beam in the Stem-Gerlach experiment is split into a certain number of separate beams, corresponding to the number of angular- momentum components along the magnetic field direction Mz=hk. 20 - 0060 306 QUANTUM MECHANICS [Part III Let us denote the largest eigenvalue quantity k by the letter 1. Then it is obvious that l^k-^ - I, (30.16) i.e., k takes on 2 Z + 1 values. The eigenfunction corresponding to Mz=hk is V (the factor — — is introduced for normalization, V 2n If each of the separate beams is once again passed through a magnet¬ ic field parallel to the z-axis, there is no further splitting; this is because Mz in these beams has a single definite value and not the whole set of values in the range hi > Mz > — hi, as was the case in the initial beam. From this the meaning of the orthogonality of eigenfunctions is very well seen. If a particle is found in a beam corre¬ sponding to a given value of k, then the probability of finding it in a beam with a different value of the projection Mz—hk'^hk is equal to zero. From the general rule, the probability is equal to the square of the modulus of the coefficient of the expansion Ck of the function {k) in terms of the functions (k'), i.e., in accordance with the gen¬ eral expression (30.11) an Ck' = J (j'* W'}' {^')<^9 • o From the orthogonality condition (30.6), the integral is naturally equal to zero ySk'^k. Therefore the orthogonality condition is a neces¬ sary condition of particles being found in states with definite values of Mz or, as in the case of an arbitrary operator X, in states with defi¬ nite values of X. But the orthogonality condition follows directly from the Hermitian nature of operators (30.3), while equation (30.2), concerning the functions with equal values of X, is insufficient. The Hermitian condition implies the reality of eigenvalues together with the possibility of “pure states,” i.e., states with definite eigen¬ values of quantities. If the second magnetic field is along the a;-axis, then splitting will again occur due to the component of angular momentum Mx, which does not exist simultaneously with The number of splitting com¬ ponents is again equal to 2 Z -f 1, since it is determined by the maximum angular-momentum projection 1. This quantity cannot depend upon the direction of the magnetic field, and is related only to the atomic states in the original beam. (30.16) 2Tt ||4-N9=i). Sec. 30] EXPANSIONS INTO WAVE FUNCTIONS 307 The eigenfunctions of Mx are = (30.17) V 2tc where — I, and w is the angle of rotation about the a:-axis. Functions (30.16) and (30.17) do not coincide, which is a natural consequence of their being functions of noncommutative operators. As a result of magnetic splitting in a field directed along the a:-axis, a beam with given value of k is split into 2 Z +1 beams with definite values ki- Hence, the function (30.16) will be represented as the super- imposition of functions (30.17): I 4>(k)=2^c^,'P(k,). (30.18) The square of the modulus I c*, is proportional to the intensity of the beam of the given projection Mx = h,\ obtained as a result of the secondary splitting of the beam with a given ilf^. Averages in quantum mechanics. Let us now find the average of X in a state given by the wave function ij' (x), represented in the form of a sum (30.8). By definition the mean value is \=2;xw^, (30.19) i.e., the sum of possible values of X multiplied by the corresponding probabilities. Let us substitute here wxfrom (30.13) and cjfrom (30.11): X =27 I Cx I =“ =27 =i7^ Cx /^. a:)da;. (30.22) X The sum 27®x 'i'* (^. x) = {x), since this is an equation which is a X conjugate complex of (30.8). Therefore, 20* 308 QtTAlSfTUM MECHANICS [Part in X=JtJ^(a:)X*iJ^*(a:)da: (30.23) or, from the Hermitian condition for the operator X (30.2), (30.3), X=J.{/*(a:)Xi};(a:)da:. (30.24) Thus, in order to calculate the mean value of X in a state <1' (x), it is not necessary to know the eigenvalues of X, since it is sufficient to calculate the integral (30.24). The eigenvalues of the square of the angular momentum. If tj' {x) is one of the eigenfunctions of the operator X, then the mean value X is simply reduced to this eigenvalue. Indeed, then X = ^(1^* (X, x) X(|< (X, x) da: = xjl tj; (X, a;) l^da: = X . Taking advantage of the foregoing remark, it is easy to calculate the mean value of the square of the angular momentum. First of all it may be noted that in the Stem-Gerlach experiment the mean values of the squares of all three angular-momentum pro¬ jections must be the same, because it is absolutely immaterial what the notation of the coordinate axis is along which the magnetic field is directed: Ml = Ml=Wl. (30.25) It follows that the mean value is equal to three times the mean value Ml: W =Mi + Wl + Ml = 2iWl. (30.26) In the original beam all values of Mz—hk from —hi to hi are equally probable. This means that Ml is equal to Ml -t = h^ 'L=SZ> ^ 21 + 1 3 • {21 + 1) whence W = hH (i -I- 1). hH{l+ 1) 3 (30.27) (30.28) But it was shown in Sec. 29 that M^ is commutative with Mz, so that Jf* and Mz exist in one and the same state. In the Stern-Gerlach experiment the atoms in the beam occur predominantly in the ground state. This state is characterized by a certain absolute value of the angular momentum. Therefore, the mean value of the angular momen¬ tum in such a state is equal to its eigenvalue ilf2 = = hH (l+l). (30.29) Sec. 30] EXPANSIONS INTO WAVE PUNCTIONS 309 The result (30.29) may appear somewhat surprising because the eigenvalue of the square of the angular momentum is equal not to the square of its greatest projection but to some greater amount. However, if Ml were equal to l^, i.e., its greatest value and F, then for the remaining projections there would remain an identical zero. The other projections cannot have any definite values, including zero values, at the same time as Mz^ 0. Therefore, the square of the angular momentum is somewhat greater than the square of the maxi¬ mum value of any of its projections. The only exception is when all three projections are equal to zero (see Sec. 29). Composition of angular momenta. Knowing the absolute value of the angular momentum, we can now indicate a rule for the composition of the angular momenta of two mechanical systems. Let the greatest angular-momentum projection of one system equal hl^ and that of the other system hl^; and also let Then the projection of the smaller angular momentum in the direction of the larger one is con¬ tained between hl^ and — hl^, which, when added to the larger angular momentum, yields values ranging from h{li-\-l^) to A (Z^— l^). It follows that the greatest projection of the resultant angular momentum upon any arbitrary direction in space is equal, in units of A, to Z = Zj -)- ^2 j Zj^ Z 2 — 1, Zj -[- Z 2 — 2,..., Zj — Z 2 . (30.30) The eigenvalues of the square of the sum of the angular momenta are A* (Zj -f- Zg) Z 2 -1- 1), A* (Zj + Z 2 — 1) (Zj -f- Z 2 ),..., (Z 1 -Z 2 + 1). The rule for composition of angular momenta formulated here agrees with the result that the value of a vector sum is contained between the sum and the difference of the absolute values of the vectors. Quantum equations of motion. Let us suppose that a certain opera¬ tor X is given. It is required to find the operator form of its total time derivative, i.e., X. We shall first of all determine the total derivative of the mean value X. In accordance with (30.24), for any state with wave function A, this derivative is X = da; -t-dre -f- ' Let us substitute here the derivatives —and ■— from the ct ct Schrodinger equation (24.11), whose right-hand side we shall represent as 4'> where is the Hamiltonian operator [see (29.7)]. From this, X = J-~ • 4^*)X4' da; -f ' 310 QUANTUM MECHANICS [Part III We transform the first integral on the right-hand side in with the Hermitian condition for namely accordance 41 *) (Xij^) da: = da:. We now combine all three integrals and obtain X = (1} -f 4 [i-X - X.#j) dx . (30.31) If we now define the operator X by the equality X -= J'j'* X4>da:, then we obtain the equation (30.32) ♦ 4 A A A A (30.33) The operators of linear momentum, angular momentum, and coordi¬ nate that have been employed up till now do not depend upon time explicitly. For them, only the second term of (30.33) remains: X = [jf X - X,#]. (30.34) Thus, if a given operator commutes with the Hamiltonian operator then X — 0. It is then natural to call the quantity X a quantum inte¬ gral of motion. In accordance with the general result of Sec. 29, quantum integrals of motion have a common state with energy, since their operators are commutative. We shall now find the equations of motion for the x and p* opera- tors. From (29.7), the energy operator is equal to + U. Here, only pi is noncommutative with x ; for pi we find plx — xpl = plx — pxxpx + pxxpx — xpl — = Px (VxX — X%) + (pxX — X Px) Px = -r- P* • % It follows that m ’ (30.35) i.e., the operator x is related to the operator p* by the same expression as the quantities x = Vx and p* in classical mechanics. Let us now find p*. p* does not commute with tf. The commutator of U and p* is easily evaluated: (Upx-%U)^^\{U 84. dx h dfj , Sec. 30 ] EXPANSIONS INTO WAVE EUNCTIONS 311 whence. symbolically. _ h 8 U i 8 x Hence, Upx- - PxU (30.36) Px = dO 8 x ’ (30.37) which is completely analogous to the classical relationship between the momentum derivative and the force. The quantum equations of motion (30.34) were the starting point for W. Heisenberg, who arrived at quantum mechanics independently of Schrodinger. The equivalence of both approaches was shown some¬ what later. The wave function and measurement of quantities. The probability amplitudes characterize the properties of a system in relation to the results of measuring certain quantities. If a system occurs in a state with wave function tp (x), and the quantity X is measured, then the probability of obtaining the given value of X is [see (30.11), (30.13)] 1 Cx = |J t}'* (a:) (p (X, x)dx *. For example, in the Stern-Gerlach experiment, the particles in the original beam have all angular-momentum projections between —hi and hi. Measurement results yield 2 Z-fl beams, each of them corre¬ sponding to the 2 th angular-momentum projection given by a definite value hk. However, the same measurement in a field directed along the x-axis of the original system would split the beam according to the ath angular-momentum projections. Both angular-momentum pro¬ jections do not exist simultaneously, and the initial states of the particles in the beam were identical. It follows that, as a result of measurement, the particles occur either with a definite 2 th, or with a definite xth angular-momentum projection. A measurement of a microscopic entity essentially changes the state of the latter. This is the fundamental difference between the concepts of measurement in classical and in quantum physics; a classical meas¬ urement has an infinitesimally small effect on the object being measured. As a result of measurement, the angular-momentum projections in the original beam acquire 2 i! -f 1 values, no matter how the measure¬ ment is performed. The state of these particles after measurement is essentially different and depends upon how the measurement was per¬ formed. But by performing measurements of a large number of iden¬ tical entities, we can find out in what state they were before measure¬ ment, quite independently of the method of measurement. For this reason, a quantum measurement yields physical results which are just as objective as those given by a classical measurement though, obviously, within the limits permitted by the uncertainty principle. 312 QUANTUM MECHANICS [Part III Thus, in the Stern-Gerlach experiment it appears that the particles had an absolute angular-momentum value M^=hH while the direction of the angular momentum in space was arbitrary (a non¬ polarized beam). The repeated measurement of the zth angular-momentum projec¬ tion, in the beams which had passed earlier through a field directed along the z-axis, gives a definite value of M^=h^l (Z-j-1) and a definite value of Mz—hk. Exorcises 1 ) Expand the function + = —^, in an infinitely deep rectangular y/a potential well, in terms of functions ( 25 . 12 ). V2 ” Cn ' r , , J V 2 f . 7t (n + l)x J = J (|< t|/n dx — —J sin- - - dx = V2 7 t(n-t-1) [— cos JT (n -f-1) -f-1] = V 2 X (n -f 1) [1 _(_!)«+!] . 2 ) Find the energy eigenvalues for a symmetrical quantum top. The energy of the symmetrical top is 2Ji {Ml + MD- Introducing M^, we have 2J, 2J, Substituting the eigenvalues for the angular momentum and its projections, we at last find -- 55 -( 1(1 + 1 ) Sec. 31. Motion in a Central Field The motion of an electron in a central attractive field is the princi¬ pal problem in the quantum mechanics of the atom. And it is not necessary to regard the field as strictly Coulomb in character. For example, in alkali-metal atoms, an outer electron which is bound relatively weakly to the nucleus moves in the field of the nucleus and the so-called atomic residue (i. e., all the other electrons). The charge- density distribution for these electrons possesses spherical symmetry and therefore produces a central field. We shall suppose that the poten¬ tial energy of the electron is equal to U (r), where r is the distance from the nucleus. The energy operator and the angular-momentum integral. The equa¬ tion for the energy eigenvalues of an atom (24.22) is, as usual. (31,1) Sec. 31] MOTION IN A CENTBAi FIELD 313 Here, m is the reduced mass of the nucleus and the electron, which mass is very close to the mass of the electron. Since the field is central we must pass to spherical coordinates. The Laplacian operator in spherical coordinates was obtained in Sec. 11 (11.46). Using this expression, we rewrite (31.1) explicitly: 2 m 1_ r* dr^ dr _J_^ sina Sa sin^a ^ -]r U {r)^ (31.2) The operator involving angular difierentiation is simply the square of the angular momentum introduced by us in exercise 4, Sec. 29. Therefore, equation (31.2) can also be rewritten as 2m Br^ Br 2mr'‘ ■ + U (r) ^ 4'• (31.3) It follows that the Hamiltonian operator ^ [see (29.6)] is related to the angular-momentum operator in the following way; AAA .2 A 2m r* Br Br + A/2 ™ + U{r). (31.4) Reducing to an ordinary differential equation. The operator involves only the angles 0 - and 9 and derivatives with respect to them. All the derivatives with respect to angles in the operator ^ are con¬ tained in the one term M^, while all the remaining terms involve only r and the derivative with respect to r. Consequently, the operators and iff® are commutative, since iff® commutes with any function of r and, of course, with r itself. Commutative operators have eigen¬ values in the same state. Therefore, in a central field, the square of the angular momentum and one of its projections have (together with energy) eigenvalues, which, in accordance with (30.34), are quantum integrals of motion. All the other quantities which are not integrals of motion do not exist in the same energy state (in classical mechanics they, naturally, exist but are not conserved). Thus, in equations (31.3) and (31.4), we can substitute in place of iff®its eigenvalue A®Z (Z 1) from (30.29). Then any angular dependence will be eliminated from equation (31.3) and, in place of the partial derivative with respect to r, we will get the total derivative: JHz 1 ^ „2 At 4. 2m r® dr dr ' hH(l + \) 2mr^ (j/ -h Z7(j^=(^’4'. (31.5) It is considerably more simple to solve this equation than the partial differential equation (31.2). The form of (31.6) corresponds to ( 6 . 6 ) in classical mechanics, where it was also possible to eliminate all variables except r with the aid of the angular-momentum integral. 314 QUANTUM MECHANICS [Part III Reduction to one-dimensional form. It is convenient to reduce equation (31.5) to a one-dimensional form. To do this—^the treatment is similar to that used in the problem of the propagation of spherical waves [cf. (19.6)]—we introduce the function (31.6) Without repeating the com 2 Jutations by means of which the one¬ dimensional form (19.6) was obtained, we write down the analogous equation for x - d\x 2 m dr^ 2mr^ x + Ux = ^x- (31.7) The wave function at large and small distances from the nucleus. As long as the form of U (r) has not yet been made definite, we can con¬ sider (31.7) only in two limiting cases: for very large and for very small distances from the nucleus. The field of the atomic residue is not effective at very small dis¬ tances from the nucleus, and there remains only the Coulomb rela- tionship 1 [/=- {Z is the atomic number of the element). However, if r is very small then the term is. in s-ny case, larger than the term 17di, which involves r in the denominator only in the first degree, and all the more greater than S'ii. Hence, in direct proximity to the nucleus, the wave equation is of very simple form: (31.8) In this form it is solved by the substitution X-r«, (31.9) so that a (a-1) ==.1(1-1-1). (31.10) This equation has two roots: a = 1-f 1 and a =--1. (31.11) But the second root gives = from (31.7); at the point r — 0, this function of becomes infinite for all 1. Therefore, we must discard the root a =—1 and take the relationship between (j/ and r for small r in the form = (31.12) * The result (31.12) is true for 1 = 0 as well, even though the term in this case does not exist at all and cannot exceed 17 (r) ]i. hH(l + \) 2mr^ Sec. 31] MOTION IN A CENTBAL FIELD 316 The greater the angular momentum, the higher the order of the wave-function zero at the coordinate origin. Only for i! = 0 does it remain finite close to the nucleus. This can be understood by analogy with classical mechanics: angular momentum is the product of mo¬ mentum by the “arm,” i.e., by the distance from the origin; 1 = 0 corresponds to a zero “arm” and a zero angular momentum. There¬ fore, there is a nonzero probability of finding the electron at the origin. In the old version of quantum mechanics (due to Bohr), the electron orbit with zero angular momentum passed through the nucleus. The larger angular-momentum values correspond to larger “arms” and, correspondingly, in quantum mechanics, to a smaller probability of finding an electron close to the nucleus. The behaviour of the wave function close to the origin can also be explained as follows. A centrifugal repulsive force acts on the particle; to tills force there corresponds an effective potential energy ‘ This energy limits the classically possible region of motion for small r. In quantum mechanics the particle penetrates the centrifugal barrier, though more weakly the greater r, i.e., the higher the barrier. There is no barrier for 1 = 0 and there is nothing to prevent finding the par¬ ticle at the origin. The terms ^ ^^ must be discarded for large r in the wave equation, because U (r) is assumed to be zero at infinity, U (oo)7^:0. Then the equation is also greatly simplified: d^x _ imS Its general solution appears thus; (31.13) V— 2 V — 2 m ^ X = C'ie ' ArC^e '■ '. (31.14) Positive and negative energy values. We consider two cases. Let the energy be positive, S>0. Here, x appears as follows: . ts/'imS _ . rV2 m3 X = Cie' '■ -j-Cae ’ ^ . (31.16) Both terms remain finite for any value of r. Therefore, two constants, Cy and Cg, must be retained in the solution. We came across the same situation in considering the solution of wave equation (25.33) for a potential well of finite depth. Any general solution of a second-order differential equation involves two arbitrary constants. Let us suppose that the solution (31.12), which holds for small r only, is continued into the region of large r, where it is not of the simple form r*, but nevertheless satisfies the 316 QUANTUM MECHANICS [Part HI precise equation (31.7). A certain integral ciu’ve is obtained for this equation. But any integral curve can be represented by properly choosing the constants in the general solution. As r tends to infinity this solution acquires its asymptotic form (31.15) if «f>0. The ex¬ pression (31.15) remains finite when r->oo for any constants and Cg. It follows that, for a positive energy, the wave equation always has a finite solution for any values of r. Therefore, the values for S>0 correspond to a continuous energy spectrum, since the wave function satisfies the required conditions at zero and at infinity for any > 0. In accordance with (31.15), the probability of finding an electron at infinity for r->oo does not become zero; i.e., this case corresponds to infinite motion, as in the classical problem considered in Sec. 6 (see also Sec. 25). Thus, the general rule has been confirmed that infinite motion pos.sesses a continuous energy spectrum. Now let (f<0 or —| S\. Then (31.14) must be represented thus: r V’i m i I f '\/2 m| tf | X = Cie ■ -fCge ■. (31.16) Here the first solution tends to infinity together with r and we must therefore put = 0, so that x will involve one instead of two arbitrary constants: f V'2 ml and at r = 0, which all functions with 1^0 have. The term node instead of zero is given by analogy with the nodes of a vibrating string fixed at both ends. In future we shall call Ur the degree of the polynomial and denote by the letter n the whole sum » = «, + !+!. (31.33) It is convenient to use these quantities also in the more complex cases of many-electron atoms. Even though the energy in such a case does not have the simple form (31.32), the numbers n, n,, and I are convenient for classification of the states. I is called the azimuthal quantum number. As we know, it defines the angular momentum of an electron. The following system of nota¬ tion is used in spectroscopy: the electron state with i = 0 is called the s-state and, corresponding to ( = 1, 2, 3, we have the p-, d- and /-states. There are no greater values of I in nonexcited atoms. Combining the angular momenta of separate electrons according to the rule of vector addition (30.30), we obtain the angular momentum L of the atom as a whole. The states with L~0, 1, 2, 3 are termed S, P, D, F, while states with greater L are named by subsequent letters of the Latin alphabet. k, [see Eq. (29.21)], i.e., the angular-momentum projection on some axis in units of h, is called the magnetic quantum number, since the external magnetic field is usually directed along this axis. n, is the number of wave-function zeros as related to the radius (for and oo) and is called the radial quantum number. Finally, the sum (31.33) is called the principal quantum number. In accordance with (31.32), the binding energy of an electron in a hydrogen atom is 320 QUANTUM MECHANICS [Part in me* 13,5 ev. (31.34) An analogous expression is obtained also for the positive helium ion. Apart from the difference of = 4 times, there is a more subtle difference due to the fact that the reduced mass of the helium atom differs somewhat from the reduced mass of a hydrogen atom as a result of a difference in the nuclear masses. The state with n = l is the ground state. The atom cannot emit light in this state because it is impossible to make a transition to a lower state. For more detail about radiation, see Sec. 34. The parity of a state. The state of an electron in an atom is character¬ ized by one more property, which (as opposed to energy and angular momentum) does not correspond to any classical analogue. This is the parity of a wave function with respect to coordinates. To begin with let us consider the wave function of a separate electron. The wave equation (31.1) does not change its form if we substitute x = — x' y,= — y', z — — z'. (31.36) This transformation is termed inversion: it transforms a right-handed coordinate system to a left-handed one. No rotation in space can make these systems coincide (like left-hand and right-hand gloves) (see Sec. 16), The wave equation (31.1) is linear. Therefore, if it has not changed its form, then its solution (determined by the boundary conditions within the accuracy of the constant factor) can acquire only a certain additional factor: ^(pc,y,z) = Cii{x',y',z'). (31.36) But, in principle, the primed left-handed system differs in no way from the unprimed, right-handed system. For this reason, the trans¬ formation of inversion must involve the same, transformation factor C: i^(x',y',z')=Ci^{x,y,z). (31.37) Substituting this in (31.36), we obtain ({/ {x, y, 2 ) = (J; {x, y, z ), whence C2 = l, C'=±l. (31.38) The function is termed even for (7 = 1 and odd for (7=— 1 . The eigenfunctions of a linear harmonic oscillator possessed an analogous property; here the'energy operator was also even, ^ {x) = ^ { — x), while the wave functions alternated depending upon the eigenvalue number n (i.e., they were either even or odd). Parity and orbital angular momentnm. Let us now find out what it is that determines the parity of a wave function in a central field. Sec. 31] MOTION tN A CENTRAL FIELD 321 To do this, it is convenient to utilize its form near the coordinate origin: = (31.39) In order to find the angular dependence of the wave function as well, it is sufficient to investigate it to the approximation that yields equation (31.39) since the terms U and (l)-l-S1.1 (r.) I" I >1,1 (r.) P I ri - I dFidFg, because e |t{ii, ^ (I’l)]" and e |dii, ^ (Fg)!® represent the densities of the charge distribution. The approximation consists in the fact that the effect of the interaction on the wave functions and the so-called “exchange” [see (33.32)] has not been taken into account. If the resultant moment is unity, we correspondingly obtain the other estimation: ^2 f I > 1.1 (■•illiLV (r^) I" J Iri-rd ‘ * Here the magnetic quantum numbers are equal to zero or unity. This integral is clearly different from the previous one. Thus, there appears to be interaction between the orbital angular momenta when they are to form the total angular momentum; this interaction does not involve c* in the denominator, i.e., it is electrostatic in character. In multi-electron atoms, Pauli’s prhiciple imposes definite conditions on the choice of spatial wave functions for given spins. As an example, let us consider the state with spin 3/2, which, as was just established, is possible in a system with three p-electrons. In accordance with the Pauli principle, three different spatial wave functions for the separate electrons having !:/= 1, 0 and —1 correspond to this state. The corre¬ sponding electron densities coincide in space less than, for example, in a state with magnetic quantum numbers 1, 1, 0, to which, according to the Pauli principle, there must correspond spin projections , Y in order that all three pairs of ki, ka should be different. But the less the electron wave functions coincide in space, the less the Coulomb repulsion energy between the electrons, because the mean distance between like charges is greater. For this reason, the state to which the 22 - 00«D 338 QUANTUM MECHANICS I I’art III Pauli principle assigns the greatest possible spin possesses the least repulsion energy. There are three p-electrons in the ground state of nitrogen. As was just indicated, the state with the least energy occurs when all tliree spins are parallel. The next state, for whicli the orbital angular mo¬ mentum is equal to 2 and the spin is equal to 2 ' ^ lies approximately 2.2 ev higher, while the state with orbital angular momentum I and spin -J, lies 3.8 ev higher. We can explain why a lesser energy coiTesponds to a greater result¬ ant orbital angular momentum. Wave functions, for which the orbital angular-momentum projections differ only in sign, are closer to each other than functions for which the angular-momentum pro¬ jections differ in absolute value. But those functions which corres])ond to a closer spatial electron density distribution lead to a larger re]nil- sion energy, while angular momenta in opposite directions, when summed, yield a lesser resultant angular momentum than the angidar momenta whose projections differ in magnitude also. Thus, the state with the greatest spin possesses the least energy and, for a given spin, it is the state with the greatest orbital angular momentum that has the least energy (Hund’s first rule). This is the way the orbital and spin angular momenta are com¬ bined. In calculating the electrostatic energy only, the state of the atom is defined by the absolute values of L and 8. But a magnetic interaction takes place between the resultant orbital angular mo¬ mentum and the resultant spin angular momentum of a system of electrons, analogous to that of a separate electron (.see Sec. ,32). To a first approximation, this interaction is described by the scalar product (Aj, where (w. and p,, are the magnetic moments for the orbital and spin motion of the system, and A is a factor of proportion¬ ality. The scalar product of two angular momenta assumes as many values as are possessed by the resultant angular momentum for a given absolute value of the component angular momenta (see exercise 4, Sec. 32). This is clearly shown with the aid of a so-called vector model; a triangle is constructed on the vectors L, S and J=L-|-S. In accordance with the law of composition of angular momenta (30.30), the side J can equal L-{-S, L + S —1, ..., \ L—8 \ . The energy level of an atom with given values of L and 8 is split into as many fine structure levels as can be assumed by J, i.e., 2 jS-f 1 levels if 8 is less than L, and 2 L -|-1 levels if L is less than 8. The system of levels described here occurs with the so-called Russel- Saunders normal coupling (of orbital and spin angular momenta); the energy states with different L and 8 differ considerably more than the energy states with given L and 8 but different J. The group of Sec. 33] MANY-ELECTRON SYSTEMS 339 energy levels differing only in total angular momentum J is termed a muUiplet. In heavy elements, where the spin-orbital interaction for separate electrons is great, the spin of each electron in a shell is combined with its orbital angular momentum to form a resultant angular momentum j [see (32.15)]; only then do the angular momenta j of separate elec¬ trons combine. This may be accounted for by the fact that, the relativistic effect of magnetic-moment interactions is not small com¬ pared with the energy of electrostatic repulsion between electrons in the inner regions of the atoms of heavy elements, where the electron velocity is close to the velocity of light. The type of coupling which occurs when the j of separate electrons are added, is termed j-j coupling, j-j coupling also occurs between nuclear particles as a result of the large spin-orbital interaction characteristic of nuclear forces. The spectroscopic notation for levels. In general form, the spectro¬ scopic notation for the resultant state of an atom is written thus; 2 s + irs.« } ■ The main symbol is L, i.e., the letters S, P, D, F, etc., depending upon what L is equal to: 0, 1, 2, 3, .... As a left superscript we put 2 iS-4-1. As a right subscript we put J, i.e., the vector sum of L and S from the number of the fine-structure components. Finally, the right superscripts denote an odd {u) or even (g) state, respec¬ tively. For example, the ground state of a nitrogen atom has L = 0, /S = 3/2 and is formed by three p-electrons. Hence, its spectroscopic designation is hS?/, because the total angxilar momentum can oidy equal the spin angular momentum {L = 0), and ^ 1 = 3 is an odd number. The notations for the next two states of nitrogen are 2/)" and ®P", or, if the multiplet splitting is taken into accoimt, then ®Z)?, or ®Z)?, and or , /s /t It It depending upon the resultant angular momentum J. If the ground state of the atom has L and S not equal to zero, the resultant angular momentum is determined by Hund’s second (empir¬ ical) rule: when there are less than half the possible number of elec¬ trons in a shell, the least energy corresponds to a multiplet level for which J = \L —(S I, and to that of J=L-\-S when there is more than half the possible number. Since the electron angular-momentum I can have 21-4-1 projections, and there are two values ka for each 22* 340 QUANTUM MECHANIC'S [Part III projection ki, then there can be in all 2 (2 1 + 1) electrons in a shell with given values of I and n. The total number of electrons in an atom with a given princijial quantum number n is M-l ^2(21+1) = 2 w2. (33.1) /=o The electron configuration corresponding to the least energy occurs in the ground state. It is determined by Hund’s first and second rules. The dependence of energy on the azimuthal quantum number. Before we can go over to a description of the Mendeleyev periodic .system we have to remark on the deiiendence of the energy of an elec¬ tron on the azimuthal quantum number. The energy of an electron in all atoms, except the hydrogen atom, depends upon I as well as upon n. For large I the electron is situated comparatively far away from the nucleus; in other words, it is more weakly bound to the nucleus than for small 1. For a given n, the energy of an electron is greater, the larger 1. When the field greatly differs from a Coulomb field, the de¬ pendence of energy upon I is so strong that an increase in the princi¬ pal quantum number n, with a simultaneous decrease in I, leads to a smaller energy increase than the increase of I for a given n. In other words, the state with quantum numbers w + 1, 0 can have a lower energy than the state with quantum numbers n, 1. This will become clear in the later examples. Filling the first shells. As was mentioned, the shell with n = l is filled by two electrons in the Is-state (the 1 in front denotes the quan¬ tity n). Hydrogen has one electron in this shell and helium has two. The helium shell is completely filled and has a state. The electron configuration for the ground state of a helium atom is so stable that if any other atom approaches close to it the total energy can only increase, so that repulsion forces are produced. Helium is completely inert chemically. The forces between helium atoms are small as a result of the symmetry and stability of their electron shells. Therefore, helium gas is liquified at an extremely low temperature.* After helium, the shell structure with n = 2 begins. The first electron of this shell, i.e., a 2 s-electron, appears in lithium. The two inner Is-electrons occurring in the helium configuration strongly screen the nuclear charge and, consequently, the outer electron is weakly * The condensation of helium into a liquid at low temperatures is due to the so-called Van der Waals forces, which arise out of the mutual electro¬ static polarization of approaching atoms. Tliese forces act at larger distances than the forces of chemical affinity, and are very small compared with them. Sec. 33] MANY-ELECTRON SYSTEMS 341 bound. Such is the alkali-metal electron configuration in the case of lithium, and analogous electron configurations subsequently result each time (Na, K, Eb, Cs) from the addition of an if-electron to a nucleus surrounded by a noble-gas electronic cloud. The next 2 5-elcctron has an energy which is compai’atively close to the energy of a 2 p-electron: the energy of the electron is still weakly dependent on the azimuthal quantum number since the field is approximately Coulomb. A large energy is needed for an electron to go from a 1 s-shell to a 2 5- or a 2 p-shell, while a small energy is needed for the transition from a 2 5- to a 2 p-shell. For this reason, the beryllium electronic configuration, having two 2 s-electrons, is not very stable with respect to an electron transition to the 2 ^j-shell. In other words, filling the 2 s-shell does not give the electron configuration of a noble gas. Indeed, as we know, beryllinm is a metal. After beryllium, the 2 p-shell fills up, and is completely filled for the noble gas neon. Neon follows fluorine, which requires one electron for the shell to be filled. The energy required for an electron to be added to the fluorine 2 p-sholl, to fill the shell of neon, is large. This explains the chemical activity of fluorine and the other halogens, which are similarly situated with respect to the noble gases. There can be eight elecc-rons in a sliell for which n = 2. This is the first group of the Mendeleyev system. The shell with n = 3 is then filled, though initially only the first two subshclls: 3 s and 3 p. The elements of the second group have an outer electron-shell structure similar to the elements of the first group. The chemical properties of atoms are basically determined by the outer shells. This explains the similarity of chemical properties, on the basis of which Mendeleyev formulated his law. Argon has a filled shell, i.e., still another group of eight elements is completed. The noble-gas configuratioit is obtained for argon because the 3 p-state, on the one hand, and the 3 d and 4 s states, on the other hand, differ considerably in energy. By considering the possible states of shells which, to be filled, lack less than half the possible number of electrons, we can consider that unfilled states behave like electrons. For example, if there are two of the six electrons wanting in a 2 p-shell, then we can combine the states of the two “holes,” similar to the way that the states of two 2 p-electrons were combined at the beginning of this section. In doing so, correct results are always obtained, provided that Hund’s second rule is used in finding the total angular momentum J of the ground state, i.e., that we take J=L+S. It is easy to see that four electrons in the shell are equivalent to two holes by applying the Pauli principle first to an electron and then to a “hole,” (see exercise 2). Let us now give, in one table, the scheme for building up the first eighteen places in the periodic system of elements; this table shows the number of electrons having given quantum numbers. 342 QUANTUM MECHANICS [Part III Element »«. = 1, 1 = 0 n — 2, 1 = 0 II 11 II II O CO II II Ground state H 1 He 2 Li 2 1 Bo 2 2 B o 2 1 0 2 2 2 N ^ 2 3 () 2 2 4 E 2 2 5 No 2 2 6 Na 2 2 6 1 Mg 2 2 0 2 A1 2 2 6 2 1 Si 2 2 6 2 2 P 2 2 6 o 3 S 2 2 0 2 4 t'l 2 o 0 2 .5 Ar 2 2 0 2 G The filling order alter the 3/>-shell. After argon, the 45-sliell begins to fill instead of the 3rf-shell. The new group begins with the alkali metal, potassium. The sum n + i is the same for the ‘3p- and 4s-8hells and is equal to 4, while it is already greater by unity in the 3d-shell. The 4p-shell is filled after the 3(i-shell, with the same value of the sum n+Z = 5, and then the Ss-shell. It is seen that this rule is observed later on, too; the filling of the shells with the same sum n-\-l proceeds in order of increasing n. But there are certain deviations from this rule during the filling of the d- and /-shells. In the shells with n = \, 2, 3, there are altogether 2.1“-|-2.22-f + 2.3‘'* = 2-f8-l-l8 = 28 electrons. There are a fiuther eight electrons in the 45- and 4p-states, and another two electrons in the 5s-state. The 5s-state is followed by electrons with n + l = Q, where we begin with the least n, i.e., with 4d. There are 2 (4-1-1) = 10 more of these electrons. The 4d-electrons are followed by 6p-electrons, of which there are six, and then by the same simple rule we get the 6s-state. llarc-oartli elements. The next value of n-{-l = 7, the least being n — i. Hence, beginning with the 57th place (in actuality, with the Sec. 33] MANV-BLECTRON SVSTK.MS 343 58th place) the 4/-shell can begin to fill acquiring at once two 4/- electrons. This shell is already inside the atom as a result of the form of the potential distribution within the atom. The screening of the nuclear charge by atomic electrons leads, at large distances away from the nucleus, to the potential decreasing like -'j- instead 1 of — (see Sec. 44). If wc combine the potential energy of an electron, calculated Avith allowance for screening Avith the centrifugal energy, it turns out that the d- and /-states possess a minimum resultant effective potential energy deep inside the atom (see Sec. 5, footnote to p. 45). Indeed, the centrifugal energy is greater than the potential energy both close to the nucleus (with allowance made for screening) as well as far aAvay from the nucleus. Therefore, the effective poten¬ tial energy Um is jjositive for large r as well as for small r. In other words, for the d- and /-states, the Um curve goes higher for large r than for the s- and p-states, and it turns out that the effective poten¬ tial Avell for d- and /-electrons is situated closer to the nucleus than to the boundaries of the s- and p-electron shells. Thus, the d- and /-shells are, as it AA'cre, filled inside the atom. But the chemical prop¬ erties of atoms depend mainly upon the outer electrons which, in filling the 4/-shcll, change very little. This is how the group of 2 (2‘3 4 - 1) —14 chemically similar elements, termed rare-earth elements, originates. It should be ])ointed out that the d- and /-shells arc not filled succes.sively as a result of “competition” Avith outer shells: for example, there are three d-electrons and tAvo s-elocti’ons in Fjs, five d-electrons and one s-electron in the next element Avhile Mugj also has five fZ-electrons but tAvo .s-electrons. The statistical theory of the atom, which will be set out in Sec. 44, permits us, in rough outline, to find the potential distribution inside an atom. It becomes possible, from this distribution, to ])redict rather accurately the places in the periodic system Avhere elements Avith 1 = 2 and 3 appear. The 5/-shell fills up (beginning with thorium) in a Avhole group of elements similar to the rare earths. A large part of this gz’oup consists of the artificially produced transuranium elements. The wave equation of a Iwo-electrou system. We shall now formulate Pauli’s principle using wave functions. The simplest way to do this is to consider a two-electron system. The wave equation for two electrons may be Avritten thus: Here Aj and Ag are the Laplacian operators Avith respect to variables of the first and second electrons, 17(r^, r^) is the potential energy 344 QUANTUM MECUANK'S [Part III of their interaction with the external field and with each other. For example, in a helium atom U {Ti, r^) _ 2 6^ c2_ ri' ' rz + I r,- r^i ' (33.3) The wave function depends upon the spatial and spin variables of both particles: (ri, (Ti ; rj5, ua). (33.4) The interaction of spin magnetic moment with orbital motion is weak. Therefore, to a first approximation, the spin-orbital inter¬ action can be neglected in the potential energy operator. This cor¬ responds to U (rj, Ta) in equation (33.3). If the effect of siiin motion upon orbital motion is small then the probability of a certain value of spin and coordinate is equal to the product of tlic probabilities of both values, and the probability amplitude ^ also divides into a product of amjilitudes ^ (i’ll j •‘2> *^ 2 ) ~ (*’i> *’ 2 ) X ('^i> *^ 2 )' (33.5) 'I'hc probiibility amplitude of orbital motion satisfies equation (33.2), provided it does not involve the si)in operator. But even when the system is placed in an external homogeneous magnetic field II, the following operator is added to [(diH) -I- (a,H)J = III (5.1 + 5.,), (33.6) where it is taken that the c-a.xis coincides with the direction of the field (the minus sign is replaced by a })lus sign because the electronic charge is negative). The action of the operator + 5-2 on the spin function simply gives the total spin i)rojection of the system. For this reason, in the presence of an external homogeneous magnetic field, is replaced by a number which is added to the total energy of the system. The symmetry of the operator with respect to particle inter¬ change. When examining the operator in (33.2) we see that it is completely symmetrical with respect to a coordinate interchange for both particles, i.e., it does not change its form if the first electron is called the second, and the second electron the first: (Cj, Oj; r2, 02 ) •— {^2' ®2> ®i)' (33.7) But equation (33.2) is linear. Therefore, if the form does not change due to operation (3^7), then the wavm function can only be multiplied by some constant number P: ^ (^ii ®i> r2> ^2) “ FO (r2, C2’ *"1’ ®i)’ (33.8) Sec. 33] MANV-KI.ECTROX SV.STKMS 345 Because r^, er^ and Tg, are involved in the same way in all the equations, we can interchange them in (33.8) obtaining ® (*’2> *’i> ®i) ~ (r^, (Tj; Fj, o-i). (33.!)) Substituting (33.9) in (33.8), we shall have ^ (r2, <^2; Ti- <^ 1 ) = (•‘2’ <^ 2 ; To or P2=l, P=±l. (33.10) In this comparatively simple case, when there are only two ])ar- ticles, the transformation is similar to the symmetry transformation for a wave function under reflection [see (31.38)]. The commutative operator for coordinates and spin variables. We can define a coordinate commutative operator for electrons P, such that 7VP(ri,r2)=-'r(r2,r,). (3,3.11) If the wave equation is symmetric.al with respect to interchange of Fj and r 2 (without interchanging tlie s])iu variables ®2) = X{'^2-'^i) - (33.12) where the eigenvalues of are likewise equal to -b 1. We denote the set of orbital quantum numbers of the first electron by the letter (in place of n^, l^, k/^), and those of the second elec¬ tron by the letter n^. Then the orbital wave function 'F is written in more detail as 'F = 'F (ni,ri-, n2,r2). It follows from requirement (33.10) that PrT* (ni, Fi; ^ 2 , Fj) = T («!, Fa; Fj) = ± 'F (n^, Vj; n^, t ^). (33.13) The function in (33.13) with an upper sign is termed symmetric; with a lower sign, antisymmetric. The wave function of a two-electron system. Introducing, in ad¬ dition, the spin quantum numbers k„^ and which determine the form of the spin wave functions (see exercise 3, Sec. 32), we write the total wave function of a two-electron system as (Wj, Fj, A'gj) t*2> ^ 2 ) * 340 QUANTUM MECHANICS [fart lEI The total permutation of spin and spatial coordinates in this function occurs as a result of the action of the P operator, which is P=^P,Pa. (33.14) Operating with (33.14) on the function O, we have P^ /jo,, Tj, oTj j r.^, U 2 ) — O (?ii, Ajoj, ro, <^2* ^ 1 ) • (33.15) According to (33.10) tliis function is also either symmetric or antisymmetric. But it can now be seen immediately that only the antisymmetric function satislies the Pauli ])rinciple. Indeed, let the states of both electrons be identical, i.e., ni = n 2 , Av, = /ro,. Then, if the function O is antisymmetric, wo obtain PO (?ij, Aoj, r^, Oj j Uj, ^2’ ^ 2 ) ~ ^ (^i» ^2> *^ 2 »^^1’ An,, r^, Uj^) “ = — 0 {ill, ha^, I’j, rs■^ \ Mj, A’o,, r2, = “ O (Wj, A/n,, Tj, J ??j, /i'n,, r 2 , U 2 ) . (33.16) By definition, the operator P interchanges only the variables r and ( 7 , and by no means the quantum numbers n, k^- 'fhe first equation of (33.16) denotes the result of a P operation, the second takes into account the antisymmetry of the wave function, while the third is obtained from the first by permutation of all four arguments relating to the electrons. The iiossibility of such a per¬ mutation for any function is obvions, since it does not matter which particle is considered first and Avhich second when Avriting down the wave funetion: the interchange of the four values n^, ka^, rj and Uj, A’o,, rj, <73 in the last equality of (33.16) simply does not denote anything: it is immateriial which arguments are Avritten first—tho.se relating to the first electron, i.e., 7 ii, k^^, Tj. Oj, or those relating to the second electroii >ij, r.,, Uj. Hence, the function ^ (a,, Tj, (Tj; Wj. r.., 02 ) is equal to itself Avith the sign reversed, i.e., it becomes zero identically. This property is possessed only bj' an antisymmetric function and not by a symmetric function; the latter Avould become identically equal to itself. But if the antisymmetric Avave function of two elec¬ trons occurruig in identical states is identically equal to zero, the probability amplitude of this state of a system of two electrons is equal to zero for any values of the A^ariables r,, r 2 , Oj, 02 . Only an antisymmetric function is compatible with the Pauli principle. The same applies to the wave function for a many-electron system: it is antisymmetric Avith respect to a simultaneous permutation of spatial and spin variables for any electron pair. This is the generalized formulation of the Pauli principle. Sec. 33] MANY-ELKCTRON SYSTEMS 347 Particles with half-integral spin. Experiment shows that all ele¬ mentary particles with half-integral spin obey the Pauli principle: protons, neutrons, electrons, and positrons. Complex particles, consisting of an even number of elementary particles with half-integral spm, have a symmetric wave function because, for a complete inter¬ change of all variables relating to such a complex particle, we must make an even number of permutations of the elementary particle variables it consists of. But by changing the sign an even number of times we do not change it at all. For this reason, nuclei Avith even atomic weights (for example, D^, He^, O'®, etc.) and, therefore, having symmetric wave functions are not subject as units to the Pauli principle, while He®, LP, etc., have an antisymmetric wave function, that is to say, they are subject to the Pauli principle. Elementary particles not subject to the Pauli principle. Light quanta do not obey the Pauli principle since there can be an unlimited number of quanta in a state with a given wave vector b and given polarization. All particles with integral spin possess a wave function which is symmetric with respect to a complete permutation of the variables relating to any pair of particles. The Pauli principle and the limiting transition to classical theory. The Pavdi principle enables us to understand why the wave prop¬ erties of light quanta are conserved in the limiting transition to classical theory, while the wave properties of electrons are not. Wo shall consider quanta in definite states, i.e., Iraving a certain polarization and wave vector. The number of such quanta can be infinitely large, since quanta are not subject to the Pauli princi])le. We note that this was not introduced as a supplementary liypotliesis concerning the properties of light quanta, btit was directly obtained in Sec. 27 in the quantization of electromagnetic field equations: the number of quanta in a state Nii,o is the qiiantum number for the corresponding oscillator. If this quantum number is large then the motion of the oscillator becomes classical, and, as we know, its oscillation amplitude is proportional to the amplitude of a field with a given polarization and wave vector. Thus, the limiting transi¬ tion yields a classical wave pattern. In accordance Avith the Pauli principle, there caimot be more than one electron in each state. Therefore, the probability-amplitude absolute values are always limited by the normalization to unity and, consequently, do not pass to wave amplitudes which can be defined classically. i The ortho- and para-states of two electrons. Let us now return to the case for which the wave function can be represented in the form (33.5). Since the whole product is antisymmetric, one of its factors must be symmetric and the other antisymmetric. This simple result refers only to the two-elcctron problem. 348 QUANTUM MKUHANKJS [Part 111 Let us consider the wave function for two electrons. Since the spin of each electron is equal toY(in atomic units), the resultant spin can only he equal to zero or unity. Both these states of a system of two electrons have special names. The state with spin unity is termed the ortho-state, while that with spin equal to zero is called the para-state. As has already been said, the magnetic interaction of spins is small. If wo can neglect it, then it is easy to write down the spin wave functions for the ortho- and para-states. Let x (^o,; Oi) be a function of the spin variable for the first particle Si, assuming, as we know, only two values 0 ^ — 1 and == 2. denotes the eigen¬ value of the spin projection. Depcndizig upon whether is equal to or —^ , the function x has the form shown in exercise 3, Sec. 32. Without assigning a definite form to x> we Avrite down the spin wave function for two particles which do not have a spin magnetic interactiozi; X {k„,, ai-,k„^, *^1)] > XmUym ■ - J ’ ®l) (- 2 ’ ^2) ~ ^ ( 2' ’ "2) ’ (33.19) (33.20) ■ -- - is introduced for normalization. V2 The magnitude of the spin projection, i.e., 0, 1 or—1, depends upon the choice of the z-axis. But the symmetry or antisymmetry of a wave function is an internal property and cannot depend upon the choice of coordinate axes. Therefore, the state (33.19) must be regarded as one state together with (33.18), if we judge by the total spin value. They are distinguished by the spin projections, and the total number of these states is three, as is required for a total spin equal to 1. The upper line of (33.18), as is evident, corresponds to Sec. 33] MANV-ELEOTKON SY-STEMS 349 a total projection 1, the lower line to a projection —1, and (33.19) to a projection 0. (33.20) corresponds to a total spin of zero. In the accepted terminology, the state with unity spin is to be regarded as the ortho-state while that with zero spin, the para-state. This definition of ortho- and para-states from the symmetry of the spin wave function also holds for particles with spin other than . But the resultant spin in the ortho- and para-states turns out, in this case, to be ambiguously related to the symmetry of the function. The example of deuterons with spin 1 will be examined in Sec. 41. Ortho- and para-states of helium. The two electrons in the helium atom can occur either in the ortho-state or the para-state. In the first case, the atom has spin unity, in the second case, zero. The symmetric and antisymmetric spin functions are the eigenfunctions of the spin-commutation operator Pa', when operated upon by the operator they give tl, i.e., the eigenvalues of P^. To the approximation (33.2)-(33.3), the Hamiltonian* is commutative with Pa, so that Pa is an integral of motion. Therefore, transitions between the ortho- and para-states, during which the total spin is not conserved, arc far less probable than transitions with conser¬ vation of spin. Only when spin-orbital interaction is taken into account, when the wave function cannot be expressed as a product of the form (33.5), is Pa not an independent integral of motion. But the cor¬ responding terms in the Hamiltonian,** which describe the spin- orbit interaction, are inversely proportional to c*. To this approxi¬ mation, it is only the total permutation operator of the spin and spatial variables for both electrons P that is an integral of motion, because the total wave function of the two electrons is always antisymmetric in accordance with the Pauli principle. The eigenfunction of a hydrogen molecule in a zero approximation. Concluding this section we shall consider the quantum mechanical explanation for the homopolar chemical bond. Such a bond occurs, for example, between the two atoms in a hydrogen molecule. It was first considered by Heitler and London. We assume that the atoms are independent in the zero approx¬ imation. Each electron is situated close to its own nucleus. We shall denote the nuclei by the letters a and 6, and the electrons by the numbers 1 and 2. In the initial approximation, the interaction ♦ The Hamiltonian moans tho Hamiltonian operator. ** Seo A. I. Akhiezer and V. B. Berestetsky, Quantum Electrodynamics, GTTI, 1953, equation (37.10). [English translation by Consultants Bureau, Inc. New York, N. Y., 1957.] 350 QUANTUM MECHANICS [Part III between atoms is not taken into account. But this does not mean that the wave functions of two electrons can be taken in the form 'F = tj; (r«.) (rt,), because this function is neitlicr symmetric nor antisymmetric with respect to the interchange of the electron coordinates. Neglecting the spin-orbital interaction, we must write the spatial wave function in one of the two forms: T = ^ (r,.) {r,,) + ^ (r,.) 4 , (r^.), (33.21) or = 4- (^ 0 .) 4- (^0 - 4; (rt.), (33.22) assuming that the total wave fimction O is obtained by multii)lication of the spatial function by the spin function of opposite symmetry. In this form the wave functions are compatible with Pauli’s principle. VVe ivrite r„^ and in scalar form because the wave functions of a hydrogen atom in the giuund state do not depend upon the angle. The wave equation for a hydrogen molecule has the following form: I--—_ ^ -^ — ^- \ imp imp im im ^ ra^ rb, - -T = .^T. (33.23) The first two terms describe the motion of the nuclei of the molecule. They involve the mass of the proton mp in the denominator, and are therefore exceedingly small compared with the terms describing the motion of the electrons. Physically, this means that the nuclei move considerably slower than the electrons, so that we can find the electron wave function for a fixed distance between the nuclei. Then if is a function of the distance between the nuclei. If this func¬ tion has a minimum, corresponding to a stable equilibrium for a given electronic state, then it becomes possible for atoms to form a molecule. Wo shall not in future write the terms corresponding to the nuclear kinetic energy ; they must be taken into account when we consider the vibrational, rotational or translational motion of molecules, though the very position of stable equilibrium, which is determined by the electronic motion, can be found without allow¬ ance for —— Aa and — Ai,. imp imp The terms of the Hamiltonian appearing in the first line of (33.23) without — Aa and —^^^Ai,) where the index 0 indicates the degree of tTlp f ^ the approximation. The second line involves terms due to atomic Sec. 3:J] MANV-ELECTBON SYSTEMS 351 interactions: the attraction of electrons to “alien” nuclei, and the Coulomb repulsion between electrons and nuclei. We shall call this part and consider it a perturbation: this is true, strictly speaking, only in a qualitative sense. Perturbation method. Using the notation and the wave equation is written as = (.^0 + jTi) T = 6' or 'Bg, and two states with imity spin: ap, 5P or sp^, apf, ®Pg and ^Ff^. 2) Show that, in a system of four p-electrons with tho same principal num¬ bers, tho states are tho same as in a system of two p-olectrons; in other words, that two electrons have tho same states as two “holes.” Sec. 34. The Quantum Theory ot Radiation In this section we shall find the probability of an excited atom emitting a light quantum in unit time, and we shall compare the probabilities of such radiation transition.s as corre.spond to various changes in the atomic states. But first we must deduce a general formula for the probability of quantum transitions (this formula will also be used in Sec. 37). Transitions between states with the same energy. Let us suppose that a system has two states corresponding to the same energy but different in some other respect. For example, in this section we 23 - 0060 354 ejUAN'J'UM WKCJHANICS [Part III will coiisider an excited atom having energy excess — (^o above the ground state. This atom is capable of emitting a light quantum with energy Aoi =<$’, — in the excited state Af is such that is con- siderably less than the energy level spacing of the atom, then the energy uncertainty can be neglected to a first approximation, assuming that the atom initially occuiTed in a state with an accurate energy value it is also necessary to calculate the probability that, in a certain interval of time t, the atom will go to the gi'ound state, and a quantum with energy h co — Sq will appear in the electromagnetic field. The reason for the traTisition is interaction with the electromagnetic field. Here the lifetime of the atom in the excited state is so great that — (jfj. For this reason, the interaction of the atom with the electromagnetic field can be interijreted as a small pertur¬ bation superimposed on the excited atom with energy S’l. The same type of problem concerning the transition probability due to a perturbation can also bo formulated for other transitions. For example, if the total excitation energy of an atom is greater than its ionization energy, it is possible for an electron to be emitted from the atom without radiation. In this case the excited state of the atom and the ion-]-electron state belong to the same energy. Each of them separately does not have a strictly defined energy. Transition probability. A radiation transition with the emission of a quantum is caused by interaction between an atom and an electromagnetic field. We shall suppose for the time being that this interaction is “switched off”; then the energy of the atom and field separately becomes an exact integral of motion. We shall call its eigenvalue in the initial state tfj. Then, if the interaction is “turned on,” a finite probability exists of the system making a transition to some state which, energetically, is very close to but otherwise very difl'erent from the initial state; for example, the atomlwas excited in the initial state and there were no quanta in the field. Soc. 34] THE QUANTUM THEORY OP BAtlTATION 355 while in the final state the atom went to the ground state and a quantum appeared in the field. Let us divide the Hamiltonian of the system into two terms: , where corresponds to the separated atom and field while describes the interaction. We then deduce a general formula for the transition probability, and apply it to a radiation. Wo shall therefore call the Hamiltonian of the unperturbed system, and regard jf’d) as a small perturbation causing the transition. The eigenfunctions and eigenvalues of the operator are deter¬ mined from the equation • (34.1) Allowing for perturbation, the wave function satisfies the equation “ T 4r = (3^-3) Considering that is a small perturbation, we represent the wave function in the form = (34.3) the “product” .3^^® ij ^13 > • • • t , . ^21 > ^22 > ^23 > • • • . Xjfc, . ^31) ^32 > ^33 > • • • . X3*!, . Xni, Xn2, X«3, . . • > Xfife, . . (34.13) Such a table is termed a matrix in mathematics, while the separate quantities Xhi- are called matrix elements. The right-hand side of (34.7) contains the matrix element We note an important property of the matrix elements of Hermitian operators. In accordance with the Hermitian condition (30.3) , (34.14) where the conjugate sign* on the right refers to the whole integral. I’roeeeding now from the definition (34.12), wo write: X,.fc = Xfc„. (34.1.5) A matrix whose elements satisfy equation (34.15) is termed Hermitian. The relationship between matrix elements of different quantities. Let us take the matrix elements of both sides of the operator equalities (30.35) and (30.37), and put the time derivative before the integral: 'd( ’ (34.16) (34.17) The time dependence of the matrix elements was found in equation (34.8), namely I (»r« - IA’L'IO.V 350 For clarity in notation we shall put the indices ^ of .Jf’W in brackets and not as subscripts, treating them as the arguments of the function, which in fact they are with respect to . We shall denote the argument of the sine by the letter t _j. 2h Passing to the integration variable ^ we obtain Ty_ ~ h J 5" (rfi+ dl. (34.26) The function ■ has a i)riucipal maximum for ? = 0. Its next maximum is already twenty times smaller. For this reason, in the integral (34.26), the main part is played by the values of ^ of the order unity. But then the instant of time t can alwavs be chosen so that —^ is considerably loss tlian 6\. In other words, it is ]3cr- missible, in the arguments of the functions S') and (S), to replace S simply by S^ and to take out the functions | (S^, S~S^)\^ aTid z(S=Si) from under the integral sign. It is shown thereby that if the time t is sufficiently long, the energies of the initial and final states S^ and S are defined so acciirately that tliey can be considered simply equal to one another, in aecorclance with tlie law of conservation of energy in the transition. Naturally, the law of conservation of energy holds always, hut, for sufliciently small values of t, it is impossible to determine the energy of the final state, for the uncertainty relation (28.15) for the given case is of the form {S — Si) t~2 nh. Hence, if t tends to infinity, the precise equality S—Si is obtained. Since the function —^ - decreases rapidly with increasing the integration should be extended from —oo to oo. Since the remaining values have been taken out from under the integral sign, the integral itself can be evaluated. It is From this oo — oo w=^\ yso-) {Si,s=Si) p 2 {Si) ■ t. Then the transition probability in unit time is (34.27) (34.28) (34.29) 300 QUANTUM MECHANICS [Part III We write the second argument S to emphasize that the state ij; {S) coincides with (j; (<^y) only with respect to energy. The formula (34.29) has very many applications. The matrix clement corresponding to the emission of a quantum. With the aid of expression (34.29) it is possible to obtain rigorously an expression for radiation intensity. This result is based on quantization of the electromagnetic field performed in Sec. 27. We shall not give the other, less rigorous, result based on the analogy between classical equations and the equations for matrix elements. In order to simplify subsequent computations, we shall, from the start, take advantage of the law of conservation of energy for radiation. In considering transitions of an atom from a state with energy to a state with energy (34.32) since is equal to the energy difference between the initial and final states of the field divided by h, i.e., just equal to the frequency of the emitted quantum coj.. Substituting this into the expression for the matrix element (Ak)oi we find that (A£)oi = 2]/'^“- <■£ , (34.33) since the coefficient of becomes zero. For simplicity we shall temporarily omit the indices k and a. We must evaluate the integral Qoi-j^iQwlQ- (34.34) Here, the field-oscillator wave functions are Q* — - \2 ^ ^ ]/ t c - d.c = g‘ ffi ■If 4^01 ^10 ~ fc>icr2o, (34.40) ^ ^ because, according to the law of conservation of energy, Wm = ^^ is equal to the frequency of the radiated light. The square of the modulus of the matrix element is 1.3fmi2=e=>-^^|e£r2„12. (34.41) We shall now take into account the fact that an emitted quantum can have two different polarizations. If we are not specially interested in the probability of quantum emission with a given polarization, then the probability must be summed over the polarizations, i.e., over o. To begin with, let us assume the vector k to be in the direction of the 2 -axis. Then the unit vector ej can have two directions: along the a;-axis and along the y-axis. Accordingly, {Sec. 34J THK QUANTUM THEORY OF RADIATION 303 ^10 i ' + |yiol (34.42) Let us find the average of this expression over all possible directions of quantum emission. It is then obvious that l^iol^= |yiol““ I ^10 I * = "jj’1 *"10 1^ • (34.43) By iierformhig this averaging after summation with respcot to o, we obtain O (34.44) In order to find the probability of emission of a quantum in unit 27C time, wo must multiply (34.44) by -^'-z(^). z (S') is found from equation (25.24), where Ave must put dN{<„) (34.45) Fmally, from equation (34.29), we find the exjjression for the probability to a dipole approximation; dW_ dt 3 hc^ i-e‘ * 101 (34.46) Wo can UTite the product er^g as i.e., as the dipole moment matrix element. The intensity of radiation is equal to the radiation probability in unit time multiplied by the energy of the quantum: di'i A A'. . 4'(* = /(*! (r> , because and c'*' '’’ are eigenfunctions of the angular-momentum z-projection. Hence 2it Zjo = J /o {r> h) r cos h • /j (/•>>) sin 9- dr dhj ~ > d cp ; u J (> i(p (fc - k') (l(p 0 eiolarized in the a:«/-plane can only be emitted if the magnetic quantum number changes by + 1. The rules that determine what change of quantum number governs the emission of a given radiation are called selection rules. Tlie selection rules for dijiole radiation with respect to the magnetic quantum number forbid the changing of k by more than unity. The selection rules for the azimuthal quantum number and parity. The magnetic quantum number is the angular-momentum projection. Since the angular-momentum projection does not change by more than unity, the angular momentum itself (i.e., the azimuthal quantum number) cannot change by more than unity. But I for a separate electron cannot remain unchanged, because then the functions and Aq must have the same parity. Here, the product ipo 2 will turn out to be an odd function while its integral, i.e., the matrix element | iLo zijii d F, will become identically zero. In exactly the same way V and J d F will also become zero. This is why, for a dipole transition of one electron in the atom, the azimuthal quantum number changes by ±1. Angular momentum and parity of a light quantum. As was indicated in Sec. 13, an electromagnetic field possesses angular momentum. If from equation (13.28) we determine the angular momentum of a quantum emitted during dipole radiation, it comes out equal to unity. And the state of the quantum is odd because it is determined by the parity of the dipole-moment vector components d, which, obviously, change sign for the interchange x-^ — x, y^ — y, — z. Hence, the selection rules for the azimuthal quantum number and parity of the state of the atom must be interpreted as the conservation laws of total angular momentum and total parity of the atom-)-quantum system in radiation. Clearly, if the angular momentum of a quantum QUANTUM MECHANICS [Part III 3(i6 is equal to unity, the angular momentum of the atom cannot change by greater than unity during radiation. The selection rules for spin and total angular momentum. If the spin is in no way related to the orbital motion, the spin functions for the initial and final states must be the same, otherwise the transition dipole moment is equal to zero due to the orthogonality of spin func¬ tions that correspond to different spin eigenvalues. This selection rule is approximate in character and is valid for light atoms. Taking into account the spin-orbital interaction, we must consider the selection rules for the total angular momentum j=M±o [see (22.15)]. 8 ince the angular momentum of a dipole quantum is equal to unity, we obtain the condition for j'—j: j'=j or j'=j±.\. Hero, the parity of the state must change. However, since the parity is not directly related to j but only to I, the transition j'=j is also possible. But the transition from = 0 to 7 ' = 0 is forbidden, because in this transition the quantum caimot acquire the angular momentum. It is necessary to note that the angular momenta for quanta of higher multipole order than dipole can only be greater than unity, so that the transition from / = 0 to f = 0 is forbidden for all approximations, and not only to the dipole approximation. The selection rules for many-electron atoms. By considering a light quantum as a particle with unity angular momentum, it is easy to obtain the selection rules also for cases when the states of more than one electron change. Neglecting the spin-orbital interaction, the selection rules are the following: S' =8, L' = L or Z('=B±1 and the parity is reversed. The transition L'—L is possible here, because, in a many-electron system, parity is not related to total angular momen¬ tum. Magnetic dipole radiation. A system of charges may radiate as a magnetic dipole as well as an electric dipole. Magnetic dipole radiation is usually related to a change of spin projection k^. Since the spin of an electron is one-half, the angular momentum of an atom changes by unity for a “flip” of the spin of an electron and for an unchanged orbital angular momentum. The moment of a magnetic dipole quantum is equal to unity just like the moment of an electric dipole quantum. But the parities of the electric and magnetic quanta are reversed. Indeed, the components of electric dipole moment change signs in an inversion of the coordinate system (31.35), while the magnetic-moment components do not change signs because the magnetic moment, like the angular momentum, is a pseudovector (see Sec. 16). As was pointed out in Sec. 19, the intensity of magnetic dipole ra¬ diation is less than the intensity of electric dipole radiation, their ratio being , where v is the charge velocity and c is the velocity of light. This ratio is about 10 -® for light elements. Sec. 34] THE QUAISTTUM THEORY OF RADIATION 367 Quadrupole radiation. In Sec. 19 it was shown that radiation is possible due to the change of quadrupole moment for the system. Here, electric quadrupole quanta occurring in an even state are radiat¬ ed because the electric quadrupole moment is an even coordinate function. The angular momentum of a quadrupole quantum is equal to two. Quadrupole radiation can occur when dipole radiation is forbidden by the selection rules. From Sec. 19, quadrupole radiation is obtained when taking into account the retardation inside the system. The order of magnitude of this retardation is determined by the ratio of the di¬ mensions of the system to the wavelength of the emitted light. There¬ fore, the probability of quadrupole radiation is less than the proba¬ bility of dipole radiation in the ratio where r is the size of the system. X~0.5x 10“* cm for visible light while the atomic dimensions are r~0.5 X 10“® cm. Therefore, in order of magnitude, the probability of a quadrupole transition is 10® times less than the probability of a dipole transition. Metastable atoms. If an atom can go from an excited state to the ground state only by means of a transition which is forbidden in dipole radiation, it remains excited considerably longer than for a dipole transition. For a strong forbiddence it may remain excited for a very long time (even on the ordinary scale, and not the atomic scale). Such an atom is termed metastable. Usually, in gases which are not highly rarefied a metastable atom gives up its excitation energy to other atoms in collisions and not by means of radiation. Radiation will then not be observed. But in highly rarefied matter, for example in the solar corona or in a gaseous nebula, the spectral lines due to the de-excitation of raetastable atoms are very bright. For example, in the spectra of nebulae, there occurs an intense magnetic dipole line of doubly-ionized oxygen atoms. Nuclear isomerism. Transitions with very large A; (up to 5) are observed in nuclei. For small excitation energies, of the order of several tens of kilovolts, metastable nuclei have very large de-excitation times—days or months. Such nuclei are called isomers with respect to the basic unexcited state of the nucleus. The phenomenon of nuclear isomerism in artificially radioactive nuclei was first discovered by I. V. Kurchatov and L. I. Rusinov (in Br®®). The totally forbidden transition. The transition from j = 0 to j' = 0, with an energy of 1,414 kev is observed in the RaC nucleus. Since the radiation in this case is completely forbidden, the nucleus simply ejects an electron from the inner atomic shell by means of an electro¬ static interaction; this may be explained as follows. If an internal nuclear rearrangement occurs, the charge distribution inside it somehow changes. For a 0->0—transition, one spherically QKANTI7M MK('HANK'S [Part Ill 3()8 symmetrical charge distribution is rearranged into another, which is also symmetrical, but •with a different radial dependence. Therefore, in accordance with Gauss’ theorem, only the electric field inside the nucleus is changed. The field outside the nucleus cannot change; for instance, it cannot radiate quanta. The wave functions of the s-states of the electrons differ from zero in the nucleus. It follows that a change of field inside the nucleus is capable of influencing an electron and im¬ parting to it an energy sufficient for ejection from the atom. In accord¬ ance with the law of conservation of energy, the electron, upon ejection from the atom, will have an energy equal to the energy differ¬ ence of both spherically symmetrical states of the nucleus minus the binding energy in the atom. It may be stated generally that the ejection of electrons from an atom shortens the lifetime of metastable isometric nuclei, since it makes transitions more possible. Sec. 35. The Atom in a Constant External Field A classical analogue. In considering the behaviour of a system of charges situated in an external magnetic field, it is very convenient to proceed from the idea of the Larmor precession of magnetic moment around the field. The only component of the angular momentum con¬ served in such precession is that directed along the field, both trans¬ versal components averaged over the precessional motion being zero. The situation in quantum mechanics is analogous, with the differ¬ ence that the projections perpendicular to the field do not exist as physical quantities. In this way a simple correspondence is established between the integrals of classical and quantum mechanics. The angu¬ lar-momentum projection on the magnetic field is such a corresponding quantity; it can be called a quantum integral of motion. f, An external magnetic field superimposed on an atom perturbs its state in a definite way. The Hamiltonian operator for such an atom may be divided into the operator for the unperturbed atom and the perturbation operator ./f <*> due to the magnetic field. Addition ol magnetic moments. Let us first of all write down the operator explicitly. It was shown in Sec. 32 that spin motion does not produce the same magnetic motion as orbital motion, namely, the magnetic moment for orbital motion is ( 35 . 1 ) and th(7 s])in magnetic moment e ■m c s. (35.2) Sec. 35] TIIK ATOM IN A CONSTANT BXTKltNAL I’lKLl) 3(5i) Therefore the total magnetic moment is (t=jior6+jisp =+ 2S). (35.3) Hence, the magnetic moments are not combined according to the same law as mechanical moments: J = L + S. (35.4) Comparing (35.3) and (36.4), we see that the magnetic moment of an atom is not proportional to its mechanical moment. In accordance with (15.35), the perturbation energy caused by the magnetic field is equal to (the moments are expressed in h units) = - (ttH) = (L 1- 28) = [ioH(.T + S). (36.5) Here is the Bohr magneton. The plus sign resulted because the charge of the electron is —e. We note that the magnetic energy in expression (16.35) was defined as a correction to the Hamil¬ tonian, i.e., to the energy expressed in terms of momenta. Therefore, in quantum theory it is directly interpreted in terms of operators. The vector model ol the atom. To a first approximation, the energy correction is equal to the mean value of the perturbing energy taken over unperturbed motion [see (33.31)]. Therefore, we first find the unperturbed state of the atom without the superimposition of a magnetic field. Let us suppose that a normal coupling exists in the atom between the total spin and the total orbital angular momentum (see Sec. 33), i.e., all the orbital angular momenta of the electrons are combined in one resultant orbital angular momentum L, and all the spin angulai' momenta are combined in one resultant spin angular momentum S. Examples of such orbital and spin angular-momentum composition were given in Sec. 33 (in the text and in the exercises). For example, in combining the angular momenta of two »p-electrons, the following states are obtained: ^D, ®P, and ^8. All these states are formed in accordance with the Pauli principle, and possess spatial wave functions of different forms. For this reason, in all three states, the energy for purely electrostatic electron interactions differs by magnitudes of the order of an atomic unit, i.e., by several electron- volts. Let us choose the ground state of these states. In accordance with Hund’s first rule, this is the state. We have not written the sub¬ script J here because it can have three values: J — 2, J — 1, and J=0. Accordingly, we have written 3 on the upper left. The states which differ oidy in J, for identical L and 8, are considerably closer to each other than the three states with differing 8 or L listed above. Let us estimate the order of magnitude for multiplet level splitting, i.e., the spacing of levels with different J. A magnetic field of moment 370 QUANTUM MECHANICS [Part III [i is of the order ~, so that the interaction energy of two moments is 2 ^ ^ 3 -. To evaluate the order of magnitude we put one Bohr magneton in place of p, i.e., 10“*°, and r~0.5 x 10 “®. This results in an inter¬ action energy of the order 10 “'®, i.e., thousandths of an electron-volt (in practice, greater multiple! splitting is observed due to larger (a and smaller effective radius values). In any case, the levels ®Pi, and ®Po, which are comparatively different from the other two levefe '£) and '/S', occur close to one another. The ®P level is split into three fine-structure levels which, in the given case, corresponds to the super¬ script 3. If P < S, the number of components of the multiple! splitting is determined by L. Each of the levels with a given J corresponds to a definite configu¬ ration of the vectors L and S. In classical theory, we would say that L and S are parallel in the state with J= 2 , antiparallel for J = 0, and perpendicular for J— 1. Of course, the latter one is not at all meaningful in quantum theory because only one angular-momentum projection exists. The projection oi 8 on L is equal to zero for J = 1, and the other projections do not exist. At the beginning of this section we indicated that, in the classical analogy, those components which are not conserved are, in some way, averaged over the Larmor precession of angular momenta and yield zero. In this case, we are not concerned with precession in an external magnetic field, but with that in the internal field of the magnetic moments themselves. Since J is an exact mtegral of motion we can, in a visual demonstration, consider that the direction of J is fixed in space, while the triangle consisting of the vectors L, S, and J processes about 3 in space. In the cases for which J = 2 and J = 0 the triangle degen¬ erates to a straight liue. Thus, to each of the multiple! levels there corresponds a definite vector model given by L, H S, and J. We note that this refers to normal _ coupling. An external magnetic field H causes the vectors L, S to process about its direction. It is most simple here to consider the two opposing limiting cases. We shall examine them. A weak external field. Let the external field be weak compared with the effective internal field that the multiple! level splitting is due to. Fig. 44 Since the Larmor precession frequency is propor¬ tional to the magnetic field, the triangle L8 J in this case rotates about the side J considerably faster than the precession about H. During the time of one rotation about H, the triangle can rotate very many times about 3 . Therefore, the coupling of the vectors L, S, and 3 in the triangle is not disrupted, as it were, due to the internal magnetic forces forming the triangle being large compared Sec. 35] THK .4TOM IN A CONSTANT EXTERNAL FIELD 371 with the external magnetic force. We have shown this idea in Fig. 44. Lot us now find the correction due to the magnetic field. In calculat¬ ing the mean value of the perturbation energy from the unperturbed motion, it is convenient to make use of the Larmor precession model. In this case, two forms of precession must be considered; the triangle LSJ about J and the precession of J about the magnetic field. Equation (35.5) involves the vectors J and S. It is very simple to average J: we must take its projection on the magnetic field Jz. We shall consider that the z-axis coincides with the direction of H. The projection of S upon H is not meaningful because the vector S together with the triangle LSJ rotates considerably more rapidly about J than about H. The component S, perpendicular to J, is averaged by the precession in motion unperturbed by the external field. There remains the projection parallel to J and equal to S; = . (35.6) (S J) Obviously, the projection of this vector upon H is equal to Jz . Thus, the mean value of , 5 ^ 6 ) is proportional to Jz and is equal to ^(1) = .5^, = J. (1 + . (35.7) It is now necessary to give a quantum meaning to the product (SJ). From the definition of J (35.4) we have L=J-S. (35.8) Squaring this equation, we get L2 = J2-hS2-2(SJ). (35.9) Expressing the square of the angular momentum in A units, in accord¬ ance with (30.29), we have L^=L(L + l). Let us make similar substitutions for and S®. Therefore (S J ) _ J {J 1) -f- (i) = J’(i) = iL,HJz[l + , (36.11) Thus, the fine-structure level with a given J is split into as many levels as there are different projections of J on the magnetic field, (;i:AN'rt'M MliClIAMf'S ii’lu t m :J72 i.e., 2 J +1 levels. For given L and S, the following definite factor corresponds to each value of J: J(J +\) + S {S +l)-L{L+ 1) 2J(J+1) It is called the Lande factor. For example, for L — we obtain , , 1 2-I-2-2 3 _ 1 . Analogously, for ,/-=2, B—L = \ . , 1 (5 I 2-2 3 2 - 6 ■= 2 - (35.12) I’lio level with J — Q does not split. Splitting in a strong field. 'J’lie representation of splitting set out here corresponds to reality only as long as the magnetic field is so weak that the spacing between the 2 7 + 1 levels of (i.^ gHJz in the magnetic field is small comiiared with that between the unsplit multi- plet levels themselves with differing J. When the splitting in the mag¬ netic field is comparable with that of the multiplet itself, or is somewhat greater, the pattern becomes more complicated, but in a strong field it once again becomes very simple. Therefore we shall consider the opposite extreme case, when the external field is strong compared with the internal field, so that the coupling between the vectors L, S, J in the triangle is disrupted. The necessity for this disruption in a sufficiently strong magnetic field can bo explained by the fact that S processes twice as fast as L. Then, from the classical analogy, each of the vectors S and L processes independently about the magnetic field, so that the correction to the energy is given by a different expression from (35.11): ^(1) = iro) = g.^eH{Lz -f 2Sz ). (35.13) Here Lz is the projection of the orbital angular momentum upon the 2 -axis, and 8z is the total-spin projection of the atom upon the same axis (in h units). Naturally, the total values of L and 8 are not changed by the magnetic field, though the distribution of levels in a strong magnetic field is not related to the multiplet structure, as was the case in a weak field, but only with the possible projections of L and S on the magnetic field. The vectors L and S process about the field far more rapidly than they process about J without the field. This is why the coupling in the triangle is disrupted. The projections 8z and Lz are changed by unity, therefore all the levels in expression (35.13) are equidistant. Of course, certain values of <^(i) may be repeated several times if the sum Lz-\-28z assumes the same value in several ways. For example, if i) = l, <9=1, Sec. 35j TUB .Vixm IN DOSSTANT KXTEUNAL FIEI.I> 373 then we get the following range of values of the sum: 1 + 2 = 3, 0 + 2 = 2, 1+0 = 1, — 1+2 = 1, 0 + 0 = 0, — 1+0= — 1, 1 — 2=—1, 0 — 2 = — 2, — 1 — 2 = — 3; there are in all seven equidistant values, and 1 and — 1 are obtained in two ways (i.e., each of them from the confluence of two levels), so that there are nine states in all. We note that in a weak field the same multiplet split thus: J = 2 into 5 levels, J = 1 into 3 levels and J = 0 did not split. As was to be expected, the total number of different states in the strong and weak fields is the same. The radiation spectrum for level splitting in a strong field. Lot us now see what .spectral lines appear when light is emitted from an atom situated in a magnetic field. To begin with, let us consider a strong field, because the pattern of the .splitting of spectral lines is simpler in this case than in that of a weak field. Both levels, upper and lower, resulting from two multiplets are split into a certain number of equidistant levels in accordance with formula (35.13). Let the radiation be observed in a direction perpendicular to the magnetic field. The radiation polarization vector is peiqicndicular to the direction of propagation, i.e., it is either directed along the magnetic field or in a third perpendicular direction, say along the x-axis (the magnetic field is along the z-axis). The selection rules for radiation polarized along z and along x are different. For polari¬ zation along the z-axis, the orbital magnetic quantum number mu.st be conserved. Bz is also conserved for all polarization in which the spin-orbital interaction is neglected. Therefore, all lines polarized along the z-axis, i.e., along the magnetic field, have the same frequency, Avhich corresponds to the energy difterence of the two initial levels prior to splitting in the magnetic field: the correction (35.13) is cancelled in calculating the difference — d’W. A wave pol¬ arized along the x-axis can be represented as the sum of two waves circularly polarized with opposite directions of polarization. The selection rule for these lines is that h can change only by + 1. (Jon- sequently, the radiation polarized along the x-axis has a frequency that differs from the initial frequency by ± • In observing the spectral lines emitted perpendicularly to a strong magnetic field, the original line is thus split into three lines separated by an interval which is equal to the Larmor frequency for the given field. If we drfil a hole in the shoe of an electromagnet it is possible to observe radiation propagated along the magnetic field. It is circularly polarized in the xy-plane. The selection rules for right- and left-hand circular polarization correspond to a change of ifc by ±1, so that there will be observed two lines spaced from the centre by ± . Thus, when the field is switched on, the original line will split into two lines separated by an interval equal to twice the Larmor frequency. 374 QUANTUM MKUHANIUS [Part 111 Exactly the same picture is found in the classical oscillatory motion of a charge situated in a magnetic field. This problem was considered in exercise 6 , Sec. 21. The effect of the splitting of spectral lines in a magnetic field was discovered by Zeeman before the quantum theory of the atom appeared. Therefore, the then accepted theoretical explanation of the Zeeman effect corresponded to the classical problem, where it was considered that the charge performs an oscillatory motion. However, in observing spectra, this classical picture applies only in strong magnetic fields such that the splitting of lines obtained is considerably greater than the spacing between multiplet levels. Under these conditions the Zeeman effect is termed normal, because outwardly it corresponds to the theoretical ideas of the time at which it was discovered. It may be noticed that a field which is strong for one multiplet can still be weak for another. Spectral-line splitting in a weak magnetic field. The Zeeman effect in a weak magnetic field is termed anomalous. A spectral pattern is obtained which is entirely different from the classical. First of all, the number of splitting components can differ from the normal. The distances between them are also quite different. As an example let us consider the anomalous Zeeman effect in the so-called X)-line doublet of sodium. This line is double without an external magnetic field. It corresponds to the two transitions 1 -> I and 3 ->^Si. The *P level has an orbital angular momen- 2 ' '2 I *2 3 turn 1 and spin - 5 -. Therefore, the resultant value of the total angular momentum J can be 1 -F - 5 ^ = and 1-^ 5 -. This is where we ^ Jt h U get the fine doublet structure of the ^P level in the absence of an external field. The level cannot split without a field because it has an orbital angular momentum of zero. The double 2)-line in the sodium spectrum arises in the transition from the doublet level to the single. According to our rough estimate of the fine-structure splitting, the difference in frequency between its components amounts to about one thousandth of the mean frequency of the doublet. The ®Pi level is lower than the ^P^ level. 2 2 Let us now calculate the Lande factor for tlu’ee levels. 1 ) ‘“P./,: L=l, S = ^l,, ?= 1 + 1 lkLll^+ V 2 • V? 2 " 1 • 2 £ 3 • J = V2, P=i, -sr = V2. <7 = 1 + 1 V2-V2 + V2-=’/2-1-2 2 £ 3 • 2 ) *P./.: Sec. 36j THE ATOM IN A CONSTANT EXTERNAL FIELD 375 J = V 2 , L = 0, 5 = 1/2. g = i + LUjillA±2hili^ _ 2 2 v.-Va In accordance with equation (35.11), we have an expression for the energy of the state in a magnetic field. For conciseness we 2 shall denote the quantity by the single letter p. Then But Jz takes on four values: 3/2, 1/2, —1/2, —3/2. Hence, in a field, the level splits into four levels, whose energy differences from 2 the central, unperturbed, state are ^a)(-3/^) = _ 2 p, ^(i)(_i/,) = _|p, ^(i)(+l/,) = |p, ^(I)(q.3/^) = 2P respectively. We obtain two energy values for the ®Px fine-structure level: 2 ^(^>(-V 2 ) = -~ 3 --p; ^"(+V 2 )=yP- And, finally, for the lower level, we get 2 ^(l)(_l/,)=_P, ^(l)(-f-V2) = P. Let us now find the spectral pattern. We start with the *Pi -> *5^ 2 2 transitions. The oscillations polarized along the field obey the se¬ lection rule AJ 2 = 0. Hence, their frequencies are shifted relative to the central position by ^( 1 ) , _ X/^) _ ^( 1 ) _ 1 /^) p + p = I p and by . V2) - , V2) - y P - P = - 4 P. Unlike the normal Zeeman effect, a double line has been obtained also for radiation polarized along the magnetic field. For perpendicular polarizations we have -V2)-^<^>{*5v., V2) = — & — &= — — S (right-handed 3 P 3 polarization). ,^'a)( 2 p./., V,)-^d)(25v,, -V2) = -|-P + P = |P (left-handed polarization). 370 QUANTUM MKCIIANICS [Part III Let us take the transition If the oscillation is polarized 2 2 along the field we once again have, of course, two lines, though with other spacing; ^(1) , _ 1/^) _ ^(1) _ 1/^) __ 2 p _ I P ^ m. . V 2 ) - . V 2 ) -1 p - p - -1 p • We have, for both circular polarizations: ^-(1) (2p,/^ , _ 3/^) „ ^( 1 ) (2,Sf./, , - V,) =. _ 2p + P - P , ('Pv., - V2) “ , V2) - - 3 p - p - - 3 p . These are the results for right-handed polarization. The corresponding rj splitting for left-handed polarization is S and ij- S. Thus, one component of the Z)-line is split into six Zeeman compo¬ nents, and the other into four. In the given case, the Zeeman effect remains anomalous as long as p is negligibly small compared with one thousandth of a volt, or the magnetic field is very much less than 5,000 CGSE units. A diagram of the splitting is shown in Fig. 45. The atom in an electric field (Stark effect). The multiplet levels for a certain total angular momen¬ tum J split in an electric field, too. We shall consider first of all the case of a weak field, when the level shift caused by the field is small compared with natural multii)let splitting. First of all, we must bear in mind that the aiigiilar-momentum projection on the electric field is determined only within the accuracy Fig- 45 of the sign, becariso the angular momentum is a pseudovoctor while the electric field is a real vector. In reversing all the coordinate signs, the angular-momentum components change sign while the electric- field components do not change sign. But since the choice of right- handed or left-handed coordinate system is arbitrary, the projections of the angular momenta on the electric field are physically deter¬ mined only to the accuracy of the sign. If J is an integer, the number See. 35J THE ATOM IN A CONSTANT EXTERNAI, EXELO 377 of its projections which differ in absolute value is equal to J +1 (0,1 ,..., while if J is a half-integral number then the total number of projections is; J -f-y |y , y, ..., . For example, H J = ~, there is only one nonnegative projection. Therefore, the state with angular momentum y is not split by an electric field, at any rate as long as the coupling between L and S is not disruiited. For comparison we note that the magnetic field splits the state Avitli into two states because the magnetic field, like the angular momentum, is a pseudovector. in a stronger electric field the couiiling between L and S is disrupted. In this case the scheme of splitting is the following. The vector L is integral. It has L + l projections on the electric field. We must project S onto its projection. But since L and S are both pseudovectors, the number of projections of S upon L is already equal to 2j6f-|-l. The only exception is when the projection of L upon the field is equal to zero. This level splits into <5 + 1 or + y levels according to S. The square-law Stark effect. The amount of splitting is determuaed by the relative shift of neighbouring levels. As was shown in Sec. 33 (33.31a), the shift of an energy level is equal to the average of the perturbation energy for unperturbed motion. Proceeding from (14.28), we have the following expression for the perturbation energy in a homogeneous electric field i^(i) =_• _ (dE). (35.14) But it is easy to see that the average of this quantity is equal to zero. Indeed, the wave function of an atomic state with given J is always odd or even (with the exception of hydrogen, see below). Therefore, the product 4'} '\>j must be even. From (30.24), the average of .^0) is equal to jFo) = - eE j . (35.15) But the integrand is an odd function, so that its integral is identically equal to zero. Level splitting is obtained only to a second approximation, if into (35.15) we substitute wave functions which have already been perturbed by the external field. This splitting is governed by a square- law field dependence. The linear Stark effect. In a hydrogen atom the electron energy depends only upon the principal quantum number n and does not depend upon 1. Therefore, the state with S’n is represented as a superposition of states with I varying from 0 to » — 1. But the wave function is even for even I, and odd for odd 1. Hence, the function 378 QUANTUM MECHANICS [Part III with § = Sn does not have a definite parity, so that the integral (36.16) does not become zero. Therefore, in the hydrogen atom we observe line splitting which depends linearly upon the electric field.* Highly excited atomic states always more or less resemble hydrogen- atom states, because the nucleus and the atomic residue act upon an electron, which has receded far from the nucleus, in a way similar to a point charge. The energies of these states depend upon I in ac¬ cordance with the expression (31.46). These states give a linear Stark effect if the perturbation produced by the field shifts the levels more strongly than they are split in 1. Ionization of the atom by a constant field. A constant electric field not only shifts the energy levels of an atom, but also qualitatively changes its whole state. Let us write down the potential energy of an electron in an atom situated in an external electric field E which is directed along the z-axis: U =U^(r) +eEz. (36.16) For a sufficiently large and negative z the potential energy far away from the atom is less than in the atom. The potential well in the atom is separated from the region of large negative z (where the potential energy can be still less) by a potential barrier. But there is always the probability of a spontaneous electron transition through the potential barrier into the free state. Transitions of this type were considered in Sec. 28 as applied to alpha disintegration. Any state of an atom put in a constant electric field may be ionized, but, naturally, if the field is weak the probability of ionization be¬ comes vanishingly small. In a strong field the potential barrier be¬ comes transparent, especially for highly excited atomic states. If the time for the spontaneous ejection of an electron in such a state turns out to be less than the radiation time, the corresponding line in the spectrum disappears. Thus, a weak perturbation inside an atom (the atomic imit of field intensity £! == 5.13-10® v/cm, so that the external field is always small compared with the atomic field) essentially affects the state since the conditions at infinity change. But if the broadening of the atomic levels is stUl small compared with the distance between them, they can be regarded, as before, as discrete. Exercise Construct a diagram for the splitting of the multiplet and the transitions in a strong and weak magnetic field. * The relativistic expression (38.28) for the energy of a hydrogen atom involves n and j. The orbital angular momentum 1 = / ± Va for a given j, so that a state with given n and j (in the same way as to a nonrelativistio approximation) does not have definite parity and yields a linear Stark effect. Sec. 36] QUANTUM THEORY OF DISPERSION 379 Sec. 36. Quantum Theory o! Dispersion The classical theory of dispersion, a brief outline of which was given in exercise 19, Sec. 16, proceeds from the concept of a charge elastically bound in an atom. The forced oscillations of these charges under the action of a sinusoidally varying field lead to an electrical polarization of the medium proportional to the field. Whence the dielectric constant can be easily calculated as a function of the fre¬ quency. The classical theory of dispersion is in good agreement with ex¬ periment. Yet, at the present time it is well known that the charges in atoms are by no means bound by elastic forces. For this reason, the success of classical dispersion theory may appear incomprehensible. Even though the charges are not bound by elastic forces, there exist quantities, relating to the motion of the charges, which vary har¬ monically with time: these are the coordinate matrix elements [see (34.18)]. Similar harmonic oscillations occur, as is well known, in the classical mechanics of elastically bound particles. The dipole moment of an atom, induced by an external alternating field, is expressed in terms of the dipole-moment matrix elements directly related to the coordinate matrix elements. In the present section, a quantum theory of dispersion will be formulated which will lead to the same expression for dielectric constant as classical theory; it will also indicate which quantities should correspond to each other in both theories. The wave equation for an atom in a given field of radiation. In order to calculate the dipole moment induced by a field, we must first of all determine the wave function of the atom in the external field. In contrast to the previous section, where the behaviour of an atom in a constant external field was studied, we shall here con¬ sider the interaction of an atom with an alternating external field which varies according to the law E —Eocoscot. (36.1) It turns out to be more convenient here to write down the field, straightway in real form instead of taking the real part of the final result in order to have a real Hamiltonian. The wavelength of a light ray is rightly considered large compared with atomic dimensions (this was confirmed by the estimate in Sec. 34), so that the field E may be considered homogeneous: its phase is constant over the whole atom. We determined the energy for a system of charges in an external homogeneous field in Sec. 14 [see (14.28) and (35.14)]. The correction to the Hamiltonian—due to a homogeneous electric field—^looks like jra) = _ (dE). (36.2) QUANTUM MECHANICS 380 ll’art III If we call the Hamiltonian of an unperturbed system then Schrodinger’s equation will be of the form - h- -If- = if + . (36.3) t’ ot Separating the wave function into an unperturbed part and a perturbation and regarding the perturbation as relatively small, we obtain an equation which wo have already used in Sec. 34: - - i't®) ii (1) .^(1) (1,(0). (36.4) Expansion in eigenfunctions. Wo seek the unknown function in the form of a wave function expansion with time-dependent co¬ efficients : (36.5) ft We obtained an equation in (34.7), Sec. 34, for the expansion coefficients - i>T d V . (36.6) The right-hand side of this equation depends upon time in a difiForent way from that in equation (34.7), because the perturbation operator uivolves time explicitly [see (36.1)]. Let us consider that the unperturbed state of the atom is its ground state, which we shall write with a subscript 0, i.e., Then there will simply bo the matrix element don on the right-hand side of (36.6) multiplied by — Eo cos wf. The time dependence for the matrix element was found in Sec. 34. Using the notation (34.20) we can write don — e' “«o d'o„ . (36.7) The representation of a matrix element together with its time de¬ pendence is termed the Heisenberg representation, and that without the dependence, in the form d'on, is the Schrodinger representation. Substituting (36.7) in (36.6) we obtain ” T ''' In order to integrate this equation we must impose a certain initial condition upon c„. It is natural to suppose that the external field acts for a sufficiently long time so that all the transition processes related to turning on the field do not affect the states. We can assume, for example, that the external field depends upon time according to the law: Soc. :!t)j IJl' VN'i'UM THEOUV or DISPERSIOE 381 E = Eo e“'cos 6>1 for t <0 , E = Eo cos bit for < ^ 0 , (36.9) i.e., the amplitude gradually rises with time to the value Eq. This law for the change of the field must be substituted into (36.8), inte¬ gration performed from — oo to any t, and a must tend to zero. After this, at each instant of time {t<0 or t^O) there will be a single dependence of c„ upon t: Ch 1 ~2h (w„o + w) t “n 0 + “ j (Eo d'on) . (36.10) Induced dipole moment. The mean value of the dipole moment is calculated according to the general formula (30.24) for mean values; “d =-J (A(«)*-l-ij;(i)*)d(4-(»)-)-4/(i))dF. (36.11) The quadratic term in must, of course, be discarded, since the calculations are performed to the accuracy of terms proportional to E in the first degree. In addition, the term d(p(o)fl!F does not depend at all upon E and, therefore, is irrelevant to the problem of polarization produced bj an external field. Also, this term is usually equal to zero, as indicated in the previous section in connection with the expression (35.15). Hence, the mean dipole moment responsible for disi)ersion is d =-. J (tj>( 0 )*d 4 (h) -t- t}((i)*d ({;(«))dF. ( 36 . 12 ) We shall substitute here the expansion (36.5) and integrate the series term by term: d = J dC^^F -h c* Id^r a f) . (36.13) n The integrals involved here are once again dipole-moment matrix elements. Substituting their expressions from (36.7), we write the mean dijiole moment as d 2'(c„e' “«'‘‘ d'„o + d'o„). (36.14) n With the aid of expression (36.10), we finally obtain for c„: i Uli --- + (Eodo«) d- + _«:_i“L)d„„ (Eod„o)l. \ «.)„() — M W„0 -1- 0> / J (36.15) Here we could already have written d„o in place of d^o> because the time factors of d;,Q and d^n cancel. 382 QUANTUM MECHANICS [Part III Polarization. In order to calculate the polarization of atoms by a light-wave field, it is sufficient to know only the dipole moment projection on the field. If, for example, the electric field of an incident wave is directed along the a;-axis, then the expression (36.15) involves only angular-momentum transition components directed along the a;-axis, i.e., the matrix elements of x\ n { — 1 - o> u„o + CO + "nO ' “nO + w ' 1- a:on I ® + (36.16) We have made use of the Hermitian nature of matrix elements expressed by the relationship (34.15). In other words, we have put Ixq,. |2 in place of XgnXng. Now, by performing a simple algebraical transformation and introducing the electric field E itself instead of its amplitude Ef^, we have: 1 _ 2 Ci>no 6^ [ I ^ E. (36.17) The dispersion formula. Let us consider the polarization of a medium P=Nd, where N is the number of atoms in unit volume.* The electric induction is related to the electric field and polarization by the relation¬ ship (16.23) which, in the given case, is of the form D-=E-|-47tP = (l(36.18) n But I)=eE from the definition of dielectric constant, so that s = 1 -p ' • (36.19) We note that this expression is correct only when the frequency of the incident radiation is not close to one of the natural frequencies of the atom '^IqX — ^ Ctn'l'n • (36.29) n n The expansion coefficients are determined from (30.11): a„ - I V; a,:, = j V • (36.30) In other words, they are equal to the matrix elements x^n. Now, substituting the expansion (36.29) into (36.28), we obtain ^ j"■ *"oj “ J • (36.31) n But this expression contains the momentum matrix elements, which can be replaced by coordinate matrix elements by equation (34.21): J '^^Px^nd V — iPx)nQ “ i'W6>Q/iXnQ , J V ~ (Pa')o» “ imoitiQXQtt . After this substitution, equality (36.31) can be easily reduced to the required form (36.26) if we take advantage of the fact that Xno =xon and chqii ” — oibq. We note that the oscillator fractions /,, (they are also called “oscilla¬ tor forces”) are proportional to the same matrix elements as are involved in the probabilities of radiation or absorption of the appropriate quanta. Therefore, tlie dispersion properties of a substance may be associated with the intensity of the spectral lines emitted by it. Incoherent scattering. In addition to the dipole moment d deter¬ mined by equation (36.11), we can also calculate the transition moments corresponding to radiation with a frequency which is less than that of the incident light. In other words, we can calculate the intensity of light scattering with a change of frequency. Such scattering is termed incoherent. A very important case is when the radiation energy, which remains in the substance upon incoherent scattering, contributes to exciting Soo. 37] QUAKTUM THKORY OF SCAXTKRINO 385 the oscillatory motion of the molecules. This phenomenon was dis¬ covered by L. 1. Mandelshtam together with G. S. Landsberg and, independently, by Raman. It is frequently accompanied by the exci¬ tation of oscillations which do not manifest themselves in the direct absorption of quanta as a result of the appropriate selection rules for molecular oscillation^. In this case, incoherent scattering yields im¬ portant information concerning the molecular structure of substances. Sec. 37. Quantum Theory of Scattering The effective cross-section concept in quantum theory. The concept of an effective scattering cross-section of particles, which was defined in Sec. 6 in terms of classical mechanics, is directly extended to quantum mechanics. Indeed, the differential effective scattering cross-section of the particles inside a given solid angle is the ratio of the number of scattered particles in this element of angle to the flux density of the incident particles. Since flux and flux density can be defined quantum-mechanically, the effective cross-section has the same sense in quantum theory as it has in classical theory. In practice, however, it is very difficult to calculate the effective cross-section. Therefore, we shall consider certain special cases in which the solution to the problem is comparatively simple. The Bom approximation. Let us suppose that a particle with energy S is scattered in a given potential field U. We shall first consider the case of S’ p U. Then, the change in the wave vector of the particle in the field is of the order ■\/2m(S—U) ‘\/2m S 1 /^*" ^ h “ h f 2 S T" • If the dimensions of the region in which the field acts are of the order a then the total phase change of the wave function in the scattering field is estimated as I / jn. Ua H ~2¥ ~~hr’ This quantity must be considerably smaller than unity in order that the perturbation produced by the field may be regarded as weak. In the case when TJpS, the wave number is estimated as p follows then that the criterion of smallness for a phase change is l (upper estimate). Under these conditions the action of the field TJ must be regarded as a weak perturbation imposed upon the wave function. We shall proceed from the general formula (34.29) for the transition probability. Let the initial momentum of the incident particle equal p 25 - DOSD 386 QUANTUM MJJCHANirs [Part III prior to scattering in a centro-of-mass system (see Sec. 6), and p' after scattering. We consider the scattering to be elastic, so that p — p'. To a zero approximation we choose the wave functions (p) and (p') in the form of plane waves, which corresponds to free motion. We write them as 1 JPL 1 _ ' P- C/(r)dF, (37.6) (37.7) Scattering by a central field. Simplifications appear in expression (37.6) if the field U is central, i.e., if it depends only upon r. Let us calculate C/tt' for this case. In defining the polar angle 3- we choose the direction of the vector k — k' as the polar axis. Then OO TC Uul' = Je'C^-kOr{7(;.)dF=2iT:|r2dr(7(r)|c''Ik-k'Ircos»ginda-. (37.8) 0 0 Bearing in mind that sin 3 d3= — d cos 3, we can integrate with respect to 3 immediately, obtaining Uul' = ‘2njr^dr U (r) | i I k — k' r — r C7 (r) sin ( 1 k - k' 1 r) d r . (37.9) As we have already said, k = k'. Therefore, the vector difference is easily expressed in terms of the deflection angle 6 for the particle: |k- k'|2 = 2)fc2- 2(kk') = 2ifc*(l - cos 6 ) = 4A:2sin2 |-. (37.10) This can also be seen from a geometrical construction. We have = fcsin OO —^ Cr U (r) sin l2kr sin —j dr. smy J (37.11) Thus, a calculation of Utic' reduces to calculation of a single integral (37.11). Rutherford’s formula. For the case of a (k)ulomb field, U = ±-. The integral Uu^' is found in the following artificial manner. We define the integral J sin x dx thus OO lim fsin xdx— lim -5-^1 = 1 25* 388 QUANTUM MECHANICS [Part ni Then and, finally, Jsinascda: = U. kk' OO 271 ^ 6 ” r . I 2fcrsin 1 a ± TzZe^ yfc'isin^-i (37.12) Substituting this in (37.7), we obtain a final expression for the differential effective scattering cross-section: da = Z^e*dCl 0 ^ 4 OT* v* sin‘ - - (37.13) where we have taken advantage of the fact that p = hk~7tiv. This result curiously agrees with the precise classical Rutherford formula (6.19). It turns out that the result (37.13) is also obtained from a precise solution of the wave equation for the case of a Coulomb field. Thus, Rutherford’s formula is extended to quantum mechanics unchanged. The Born approximation in the theory of scattering by a Coulomb field can be regarded as a series expansion in square powers of the charge, or, more precisely, Ze®. But since the precise formula does not involve powers higher than the result of the Born approxi¬ mation coincided with the precise result. We shall now estimate the limits of applicability of the method under consideration for the Coulomb field. To do this, we make use of the first criterion established at the beginning of the section for the applicability of this method. Since the product Ua in this case is equal to Ze^, we arrive at the following condition: Ua h Ze^ . - (37.14) The quantity e®/Ac = 1/137. Therefore, we write (37.14) otherwise thus: 137 V (37.16) But Z~90 for heavy elements, so that (37.15) is not satisfied in general. Of course, Rutherford’s formula is applicable to nonrela- tivistic particles in this case too, because it is exact; but in calculating a correction, for example, arising from a distortion of the nuclear field by the field of atomic electrons, the Born approximation yields an incorrect result. Sec. 37] QUANTUM THKOEY OF SCATTKRINO 38S) With the condition (37.15), formula (37.13) is applicable also to the scattering of relativistic particles at small angles provided m is re¬ placed by The collision parameter (aiming distance) and angular momentum. The Born approximation cannot be used when large forces act upon a particle, even if they are concentrated in a small region. First of all let us define what is meant by a “small” region. ft is convenient here to compare the classical aiming distance of p (see Sec. 6) with the angular-momentum eigenvalue in quantum mechanics. For large angular-momentum eigenvalues, when the quasi classical approximation is applicable, we can give the following esti¬ mate ; Whence hi ~ mv p. P hi m» X/ 2tt’ ’ (37.16) (37.17) where X is the de Broglie wavelength. It can be seen from here that to a change in the angular inomeututn by unity there corresponds an increment of * 2 ™ ~ I aiming distance. Accordingly, the smallest collision parameter is given by i = 0 and p ~ - 2 "-. Here the particle is scattered in the s-state. Let us consider the case when the radius of action of the forces is less than . Then a particle with an angular momentum other than zero hardly at all experiences scattering. We have shown [see (31.12)] that the wave function for a particle with angular momentum I becomes zero, like W, at the origin. Therefore, the probability of finding a particle with (> 0 in the region of action of the forces is very small if the radius of action of the forces is much less than . Separating the wave function with zero angular momentum. Let us take the term, corresponding to 1 = 0, out of the wave function. To do this, it is necessary to expand the function (37.1), i.e., a plane wave, in a series of eigenfunctions of the operator The function corresponding to Z = 0 is especially simple: it does not depend upon the angle. Indeed, the operator involves only angular derivatives. Operating upon a function which does not depend on the angle is equivalent to multiplying the function by zero. Normalizing the angu¬ lar function of the s-state to unity, we find _l_ \/4n then J I 9 o I® dQ 1 . 90 = 390 yUAJITUM MECHANICS LPart ill The expansion coefficient for this function is, according to the gen¬ eral formula (30.11), c(r) =J(po(j;(0)(p)dF= = 1Z _ = » sin 0 do = 2 ]f^ . (37.18) V47:FJ V V kr 0 c (r) satisfies the radial wave equation for a free particle (31.6), if we put 17 = 0, 1 = 0 in it. In actual fact, equation (31.7), which is obtained from (31.5) by substituting = when 1 = 0, f7 = 0, has the solution sin-^—T— r. But — k, so that the function x = ^c (>■)• c (r) tends to a finite limit when r = 0. This corresponds to the boundary condition for a radial ij^-function at the coordinate origin. Let there now be, close to the origin, a scattering field U (r), which diminishes so rapidly with distance that U (r) = 0 when r ~ gT* in its radial dependence, the s-state wave function satisfies, as before, equations (31.5)-(31.7) for because no forces act upon the particle in this region. The solution to (31.5), which is more general than (37.18), is c' (r) = ^ - 2 ]/^ , (37.19) where 8 is some phase shift depending upon the definite form of the potential d (r). Naturally, the solution (31.19) cannot be extended to r < -s—, because the particle in this region is no longer free from the M 7C action of forces. We shall show that the effective scattering cross-section is expressed in terms of 8. An example of determining S is given in exercises 2 and 3. Determining the scattered wave. Let us suppose that, in some way, an exact solution V - we obtain = [c' (r) — c (r)] (Po sin (A;r -f 8) — sinAjrl * _ kr J "v/iTt (37.21) But for large r the scattered particles can move only away from the gifcr scatterer. This means that tj^scat involves only the function -y and g-tHtr gtfer does not contain the function ^ . Indeed, if we put il* == —p— in (24.20). then the flux j will acquire a positive sign, while if we substi- f~ikr tute - the flux will be negative, i.e., incoming. We write Ascat in complex form: =--L- [A (e‘(*'+«) - c-‘»f + '») - (e**^ - . (37.22) ZiyV ■ kr To exclude the incoming wave 6“'*", we put Ae~‘* —1, whence A=-e'«, (37.23) 1) • (37.24) 2 ^ V The effective scattering cross-section. From (24.20), the flux density of the scattered particles at infinity is The total effective scattering cross-section scattered particle is an electron. Then, strictly speaking, we should have formed a function which is antisymmetrical together with the function of the atomic electron; this we did not do. Tho final formiila for tho effective scattering ero.ss-.section differs from (37.13) by the .square of tho screening factor. We note that this factor is cori-ectly obtained only in the Born appro-viination, in contrast to Rutherford’s formida (37.13), which is exact. For 0=0, the effective cross-soction turns out to bo finite, bocauso 0=0 corresponds to large aiming distances, when tho nuclear charge is screened by the charge of an electron. 2) Calculate the effective scattering cross-section for a pai'ticlo by an im¬ permeable sphere of radius a, which is very much less than -jr— = . A 7T fC In accordance with (25.1), the wave function at tho surface of tho imper¬ meable sphere becomes zero. Hence, tho solution (37.19) has tho form c' (r) = 4 • 2 sin k (r — a) kr From this, 8 = — ka, while a k^ sin“ ka . B\it ka 1 from the conditions, so that sin ka ~ ka. Finally, cr = 4 w o*, i.e., the effective scattering radius is twice tho radius of the sphere. In classical theory a = (see exercise 1, Sec. 6). 3) Examine the scattering of particles with energy S in the s-state by a spherical potential well of constant depth | U„\ and radius a; consider that there exist the following relations: - ■ < e 1 8 ma* (1 + e), 394 QUANTUM MECHANICS [Part 111 (see See. 26). For e > 0, there exists in the well a bound state of a particle of energy close to the upper boundary of the well. Unlike Fig. 38, it is assumed here that U becomes zero for r > a. Express the cross-section in terms of the energy level ^o- The conjugation condition for the wave fimctions with r = a is of the form k cos (ka + S) _ x cos xa 8in(l:a-f8) sin xa ’ . + \U„\) , \/2m (S’ xt i *• i it • where x = — -v —■—, k = —j-. Neglecting to, wo obtain A A an expression for the effective cross-section: ire . ire ire o - am d - ^ ^ . In accordance with the conditions imposed upon Ej |f/ol wo have, approx¬ imately, cot xa ^ . 4 From (25.41) the condition for finding the level ffp is of the form Xp cot Xp o = — X Supposing that Up e . For ire '\/2 Yfh ! ^ I where lip = - -^. We note that the formula obtained also holds h> for c < 0, when in actuality there is no level at all in the well. In this case we talk about a “virtual” level. The straight line in Fig. 39 intersects the first half-cycle of the sinusoid just before xa = • A similar case occurs when neutrons are scattered by protons with anti- parallel particle spins. Sec. 38. The Relativistic Wave Equation lor an Electron The equation lor a spinless particle. Schrodinger’s equation (24.11) is formed on the basis of the nonrelativistic relationship between energy and momentum 2m + u. Therefore, it can be applied only to electrons whose velocity is consid¬ erably less than that of light, and whose kinetic energy is considerably less than the rest energy: Sec. 38J THK aELATIVISTlC WAVE EQUATION 395 r = — mc^. Immediately after Schrodinger obtained the nonrelativistic equa¬ tion, the first attempts were made to build a relativistic wave equation (Fock, Klein, Gordon). In formula (21.30) — 69)® = — h B -i- was substituted in place of as is usual in quantum mechan- t Ot ics, and in place of p, the operator V. In this way a wave equation was obtained in relativistically invariant form (t-It+ 4- + ®'?)'{' = = c® (4 V - -J A ) (i V - -J a) 4- -I- m® C-* , (38.1) which equation, however, is not applicable to electrons. The fact of the matter is that equation (38.1) does not take into account the spin of the electron, because it involves only a single wave function. Yet in Sec. 32 we saw that a particle with spin 1/2 must be described by at least two wave functions. These two functions could be introduced into the nonrelativistic equation purely formally, assuming that each of them satisfies it. But the interaction of spin and orbit is a relativ¬ istic effect; therefore, a correct equation for fast electrons must take it into account automatically, without any additional hypotheses concerning spin magnetic moment. This equation must involve op¬ erators which act upon the spin degree of freedom. The inapplicability of equation (38.1) to the electron was very quickly seen; the fine structure of the levels of the hydrogen atom obtained from this equation was incorrect. A nonspin equation cannot explain, first of aU, the number of splitting components; this is deci¬ sively against it. Charged particles without spin—mesons—^take part in nuclear interactions. Equation (38.1) can be applied to them, at least if it is shown that such mesons can be regarded, to some sort of approxi¬ mation, separately from protons and neutrons. But for electrons one has to form a relativistic wave equation that takes spin into account. Such an equation was obtained by Dirac. The Dirac equation. Following the line of Dirac’s argument, we begin with the equation for a free electron. The starting relationship is ~ Ar + m®c® = V c® (pi -f pf + p%) -b m®c*. (38.2) 390 QUANTUM MKCHANICS [Part. Ill Instead of S’ and » we must substitute the derivatives — and ^V. However, to do this it is necessary to define the meaning of the square root of an operator. Dirac supposed that, in the operator sense, a root is equal to an expression like V + p? + pi) + m^c* — c («.r p* + ay J)y + 5.zi>z) + Pmc*, (38.3) where a*, ay, a^, and p act on the internal degrees of freedom of the electron such as, for example, the spin degree of freedom. Ijet us square both sides of the equation and attempt to choose the operators a«, ocy, a^^, and P in such a way as to obtain an identity, i.e., so as to eliminate terms of the typo px py, .. •, me® px, ■ ■ ■ '■ c® (pi + p? + pi) + = c® (alpl + a|p| + a|p|) -1- + + r.^{oixS.y + aya*) pxpy -|- e®(a.^aj -|- a^a.v) pxpz + c®(aya^ f S.iS.y)pypx + (ix^ + Pa.v) px + ■h mc®(ayp + PSy) py + Ttic^ (a^p -|- Pa^-) pz ■ Henoe, the operators must be subject to the conditions a.v — oCy — aS — p® == 1 , S-xS-y -(- aya.v — a.va.; -|- S-xS-x — oLyiz |- ajdty — = a.vP + P«j.' = aj’P + Pay = a^p + pde^ — 0 . (38.4) These operator equalities greatly resemble the spin operator relation¬ ships (32.10), (32.11). It can already be seen from this that the oper¬ ators 3.x, 3y, 3z, and p at least act upon the spin degree of freedom of an electron. To the accuracy of the factor 1/4, the relations (38.4) agree with (34.11) and (32.13) for 5*, oy, and Sz. But the operators a and a are not identical. This can easily be seen by proceeding from the opposite: assume that a* = CT.v, ay = 5y, a^ = 5^. In order to obtain the wave equation we must equate the right-hand Jt 5 til side of (38.3) to-perform an inversion of the coordinate system. All momentum components will change sign so that the sign of pz will also change. But the operator a^ in front of pz, if it equals S^, should not interchange the wave-function components. Therefore, the equality between the left- and right-hand sides of the wave equation breaks down when the coordinate system is inverted; but this should not be. Therefore, a^a. The necessity for a four-component wave function. We shall be confronted with the same difficulty if we consider that the operators Sx, dcy, a* act upon the same wave functions as those appearing with Sec. 38] THE RELATIVISTIC WAVE EQUATION 397 — y ^ on the other side of the equation. Therefore, we must assume that coordinate differentiation, on the one hand, and time differentia¬ tion, on the other, are applied to different pairs of functions, of which one pair changes sign in inversion of the coordinate system while the other does not. This is sufficient to ensure invariance of the equation with respect to inversion. Thus, we shall say that the wave functions depend not only upon the spin variable a, but also upon some other internal variable p, which also takes on two values. Let us define the operators which arc completely analogous to spin operators and which act upon the variable o and upon the variable p. Bearing in mind that the factor 1 / 2 , with which the spin operators were multiplied by in Sec. 32, is no longer needed, we shall write Oj, 62 , 63 and, correspondingly, p^, pg, and pj for the variable p. These operate upon the wave function analogously: (38.5) (38.6) From formulae (32.10) and (32.11) we find the basic relations for the operators a and p: — SI = 53 = I'ySiS^—iS^'yS^Si^iSi SiS^— S^Si', SiS^— 535i;CT2a3 = — > 52I ■® 3®2 I (38.7) and similarly for p. All the operators o are commutative with all the operators p because they operate upon different variables. In order that the operators a*, ocy, ot^ together should satisfy the “anticommutative” relations with the operator as in the third line of (38.4), we arrange that all three components of a are proportional to one operator of the components of p, for example, py and ^ is simply P 3 . We notice that Py interchanges the functions while pj does not. To have the operators Six, oLy, and a^ anticommutative, as in the second line of (38.4), we put them proportional to dy, a^, 63 , respectively. Thus, *.*“Pi®i> *i'“Pi®2> **“Pi®a> P“P3" (38.8) 398 QUANTUM MJEUHANIOS [Part III It is obvious that, as a result of the commutative nature of p and o and of the definition of the operators (38.5) and (38.6), all the opera¬ tors formed in (38.8) satisfy the conditions (38.4). Thus, the wave function in the Dirac equation has four components, according to the number of values for a and p (o = l, 2; p = l, 2). For convenience in future we shall number the symbols a, p from one to four, putting '{' 3 > '{'4 columns, as in Sec. 32. Now using (38.5) and (38.6), we find out how the operators a*, S.y, 5.x, and p act upon the four-component wave function ('Ki.i)] K K2,l) <]'(2, 2)1 '{'4 ■l^ia.i) '(1,1) 'I'(1,2) '{'3 ^(1,2) — Pi ' (2, 2) '{'(2,1) 4 - (2, 2) Ml.2) ['{'(1,1) 'I'l -*'{'4 '{'3 'j'l ay4' = i'l'i ; 5z’if = -'{'4 '{'1 l-'{'2 ; p4; = '{'2 -'{'3 i-'{'4 (38.9) The choice of operators (38.8) and (38.9) is not unique. It is possible to form other operators with the same properties. For example, one could have chosen p^ instead of p^. We shall examine below (exercise 4) the implications of this fact. The Dirac equation in expanded form. Summarizing, according to (38.2), (38.3) and (38.8) Dirac’s equation can be written as — = c(6tp)t{/ -F (38.10) In accordance with (38.9), this equation must be understood as a system of four equations, which we shall write explicitly, first of aH replacing —^<]^|because is proportional to the factor lEf \ (Tipi = c (p* i}'4 — ^ Pr '!'4 + Px '{'a) + 4'i. g'^i = c(px^3-\-ipy^i — px'^i)+'rnc^^z, «f^4 = c (p*(} n° -j-n or A—> n--l-p (both are possible). The mean time of such decay is of the order of 10 “^° sec, but the A particle itself was created in a nuclear collision which lasted less than 10-®^ sec. It appears rather puzzling that a particle that can be created so quickly should disappear so slowly. It seems to contradict the general reversibility of physical laws in time. This was why the A particle came to be called “strange.” The only explanation is that both processes are of a totally different na¬ ture. The generation of a particle is due to a strong interaction, and the decay, to a weak interaction. It is enough to suppose that the A particle is always created from a nucleon accompanied by a 406 QUAimJM MECHAITICS [Part III A’-meson. This has been verified experimentally, though indirectly. Such a process does not violate the baryon conservation law. If A and K particles take part in the interaction simultaneously, it is strong, and if only one of them transforms into something else, the interaction is weak. To distinguish between these two types of interaction, Gell-Mann introduced a new characteristic of mesons and of baryons—their “strangeness,” 8. It is defined as follows: Q e -^ + + -^ Here, Q is the charge of the baryon, e is the magnitude of the elemen¬ tary charge, n is the difference between the number of baryons and antibaryons (1 for baryons, —1 for antibaryons, and 0 for mesons), and -Zz is the z-component of the isotopic spin (see Sec. 32). As all the heavy particles take part in strong interactions, a definite value of rz can be ascribed to each one of them, n must be taken equal to 1 for each baryon. For nucleons, -Zz is -f y, as they can have only two values of charge: 1 and 0 . A has no charge and can have only Xz = 0. J^has three values of charge: 1, 0 , — 1 , and equal values of t^. Lastly, S is either neutral (Ta = l/ 2 ) or negative (t*= — 1 / 2 ). Substituting these values of charge and into the definition of strangeness, we find that nucleons have 8 = 0, A and^ hyperons have / 8 =—1 and 2 has 8= — 2 . It is noteworthy that^"*" and^~ are not particle and anti¬ particle, as both have n = l. Each of them has an antibaryon, for which n =— 1 . For K- and u-mesons, n = 0 , since they are not baryons. This gives 8 =0 for TT-mesons and (S — 1 for A"-mesons. Unlike f're related as particle and antiparticle. Then a selection rule is defined: the given interaction is strong only when the resulting strangeness of all particles entering into the reaction is conserved. For instance, every interaction of nucleons and 7t-mesons only is strong (if it does not violate any other conservation laws, except strangeness). The simultaneous creation of A and K particles belongs to strong interactions also, since one of them has 8 = 1, and the other 8 = — 1 . But the spontaneous decay of the A particle into a nucleon and a 7t-meson is due to a weak interaction, because, here, the strangeness is not conserved. Transitions with A8 = 2 are forbidden more strongly than with A/S = l. That is why the 2 particle must decay first into a A or^ Sec. 38] THE RELATIVISTIC WAVE EQUATION' 407 particle, which in turn decays into ^nucleons and ix-mesons. These statements agree with the cascade nature of S decay. There is no reason as yet to attribute definite values of -r* and S to leptons, since they do not take part in strong interactions. The transition to the .nonrelativistie wave equation. It is instructive, in comparing the relativistic wave equation with Schrodinger’s equa¬ tion, to perform the limiting transition. We shall consider that the energy of the electron is positive and that its velocity v is considerably less than the velocity of light. Then differs from mc^ by a small quantity-2^. If we take and mc^

P -y a) (a,p —^ a) 4 -. (38.22) When squaring the operator |s, p —must take into account the commutation relations between the components of c and also between p and A. Taking advantage of the fact that o| — 0 “ — a* ^ , and calling p *——Ax Px- -, and analoguously for Py and Pz, we shall first of all have {S' — e. Going over to vector notation, we arrive at the nonrelativistic wave equation '1' == -^ (p - 7 - (o H) ({). (38.25) Compared with the Hamiltonian operator for a spinless particle, the Hamiltonian of an electron involves an additional term: (38.26) But since 0 is an additional mechanical moment, we see that the elec¬ tron has an additional magnetic moment {io = eh - - a me (38.27) in accordance with what was affirmed in (32.17) (a here is a dimension¬ less operator). Spin differs from orbital angular momentum in that its magnetic moment does not contain a factor 2 in the denominator. Thus, the so-called spin magnetic anomaly follows naturally from the Dirac equation. The radiation-field correction to the magnetic moment. Equation (38.27) is, of course, correct only in the nonrelativistic limit. But even ill this limit it is not completely exact. As was indicated in Sec. 27, the state of an electromagnetic field in which there are no quanta interacts with charged particles. Strictly speaking, insofar as there is an interaction between the charges and the field, the state of each separately cannot be defined with complete precision. It is therefore not surprising that any state of a field is in some way perturbed by the presence of charges, and any state of the charges is perturbed by the field. As a result of this, the magnetic moment of an electron, as is shown by the rather exact calculations of Schwinger and others, is greater than one magneton by a very small quantity, whose relative fraction is ^-^hc ' result is in complete agreement with experi¬ ment. The magnetic moments of the proton and neutron do not at all agree with the Dirac theory. For instance, on Dirac’s theory, a neutral particle (the neutron) should not have a magnetic moment at all. In actual fact, the neutron possesses a magnetic moment directed opposite to the spin. This is usually explained by the strong interaction between the nucleon and the nuclear force field or, as it is sometimes called, the meson field. There is a certain analogy here with the correction to 410 QUANTUM MECHANICS [Part ni the magnetic moment of the electron. This correction is small because the interaction constant is small. Nuclear interaction is very strong, and so the result is a large “correction,” if one can use that expression for a quantity which, in the case of a proton, is twice as great as the basic magnetic moment given by the Dirac theory. At the present time we are unable to calculate the magnetic moment of a nucleon, since no theory of nuclear forces exists. Nevertheless, the nucleon is undoubtedly to some extend a Dirac particle, as confirmed by the existence of the antiproton. Energy eigenvalues of a hydrogen atom. In accordance with the Dirac equation, the energy eigenvalues of a hydrogen atom, or of any single-electron atom, are calculated in the following way; S mc^ 1 h r aZ '12 n- hi) hi) - 1 . (38.28) where n is the principal quantum number, i.e., j is the total 6 * 1 ^ electron angular momentum, “ = -^ = • If regard a.Z as small compared with unity, then the nonrelativistic formula (31.34) results. It follows from formula (38.28) that the states 2pi/j and 2st/j, with the same »^2 and j — have the same energy. In practice, these states of the atom are somewhat split as a result of the interaction of light quanta with the groimd state of the field. The calculated splitting agrees with experiment with considerable accuracy. Exercises 1) Prove that a quantum cannot give rise to an electron-positron pair in free space in the absence of an additional external field. The conservation laws in the absence of a field are written thus: II — -t- c*p* + hut = y/rn^c* + 'p\, p -i-^n = Pi. Here, p is the electron momentum in a negative energy state, n is a unit vector in the direction of the quantum momentum, Pi is the electron momentum in a positive energy state. Substituting Pj in the first equality and squaring the left- and right-hand sides, it is easy to see that this equation is not satis¬ fied. Another method of proof is based on simple reasoning. A transition to another inertial system can always make the energy of a quantum less than 2 «tc“. A quantum cannot give rise to a pair in such a system, simply because it has insufficient energy. But what is impossible in one reference system is impossible in all systems, because the possibility or impossibility of an event does not depend upon the choice of the reference system. Sec. 38] THB BBIiATmSTIO WAVE EQUATION 411 The preceding argument no longer holds if pair production is considered close to a nucleus. Here, the nucleus is at rest in one reference system and in motion in another. Where the energy of the quantum is less than 2 mc^, the moving nucleus will “help” it to give rise to a pair. Naturally, it is in no way possible for a quantum to give rise to a pair if its energy in the rest system of the nucletis is less than 2 me*. 2) Obtain the solution to the Dirac equation for a free electron. _ I^t us equate to zero. Then the first equation of (38.11) is satisfied if we take Ac (px — ipy), <(14 = --Ac pz. The second equation of (38.11) gives ■dc*(pK + py+ Pz _ A ()i 3 = 0 or ij(« = 0 . 3) Show that from Dirac’s equation there follows a charge-conservation equation which is analogous to (24.16): -^|il4 I*. Write down equation (38.18) and its complex conjugate; multiply the first by <)/* and the second by i]/; subtract the second from the first and utilize the Hermitian nature of the operators & and p. 4) Show that if {J 2 and of the secoiul pair tj)!, but make peiTnutations only inside each pair. 0) Show that according to the Dirac equation only the sum of the orbital and spin angular momenta and not each angular momentum separately satis¬ fies the angular-momentum conservation law. The total angiilar momentum is defined as J = M -f o= [rp] -f a. Jx XPy ~ 1 /Px + Yf “* “y We calctdate the commutator with the Hamiltonian: J- ---[c(axPx + aypy «zPz} + pmc^l^l^py — i/px l “ — ^XPy- PPx-l S.vSyj [C (SxP* -t- SyPy-l- S ^ P«) H g «JC®] = Jh 0 - (•.axPy{pxX~~xpx)—Ca.yPx{pyy — ypy) + Y^px{ixax^y—axS.yix) -I- -^Py(ayaxay -a^aySy) (ajcPy — SyP* -j- p* “y — Py a.v) -0. The Hamiltonian is commutative also with the square of the total angular momentum J* - J*-!-Jy-h . The integrals of motion are ./* and Jz, and not 3* and Mz, separately. PART IV STATISTICAL PHYSICS iec. 39. The Equilibrium Distribution ol Molecules in an Ideal Gas The subject of statistical physics. The methods of quantum mechanics et out in the third part make it possible, in principle, to describe ny assembly of electrons, atoms, and molecules comprising a macro- copic body. In practice, however, even the problem of an atom with two elec- rons presents such great mathematical difficulties that nobody, so ar, has solved it completely. It is all the more impossible not only o solve but even to write down the wave equation for a macroscopic lody consisting, for example, of 10®® atoms with their electrons. Yet in large systems, we encoxmter certain general laws of motion or which it is not necessary to know the wave function of the system o describe them. Let us give one very simple example of such a law. Ve shall suppose that there is only one molecule contained in a large, ompletely empty vessel. If the motion of this molecule is not defined »eforehand, the probability of finding it in any half of the vessel is qual to 1/2. If there are two molecules in the same vessel, the prob¬ ability of finding them in the same half of the vessel simultaneously 3 equal to = 1/4. The probability of finding all of a gas, consisting •f N particles, in the same half of the vessel (if the vessel is filled with ;as) is (1/2)^'^, i.e., an unimaginably small number. On the average, here will always be an approximately equal number of molecules in lach half of the vessel. The greater the number of molecules forming he gas, the closer to unity will be the ratio of the number of molecules n both halves of the vessel, no matter at what time they are (bserved. This approximate equality for the number of molecules in equal volumes of the same vessel gives an almost obvious example of a sta- istical law applicable only to a large assembly of objects. In addition o a spatial distribution, molecules possess a definite velocity distri- 414 STATISTICAL PHYSICS [Part rV bution, which, however, can in no way be uniform (if only because the probability of an infinitely large velocity is equal to zero). Statistical physics studies the laws governing the motion of large assemblies of electrons, atoms, quanta, molecules, etc. The problem of the velocity distribution of gas molecules is one of the simplest that is solved by the methods of statistical physics. Statistical physics introduces a series of new quantities, which can¬ not be defined in terms of single-body dynamics or the d 3 mamics of a small number of bodies. An example of such a statistical quantity is temperature, which turns out to be closely related to the mean energy of a gas molecule. If a gas is confined only to one half of a vessel, and the barrier dividing the vessel is then removed, the gas will itself uniformly fill both halves. Similarly, if the velocity distribution of the molecules is disturbed in some way, then, as a result of collisions between the molecules there will be established a very definite statistical distri¬ bution, which, for constant external conditions, will be maintained approximately for an indefinitely long time. This example involving collisions shows that regularity in statistics arises not only because a large assembly of objects is taken, but also because they interact. The statistical law in qnantnm mechanics. Quantum mechanics also describes statistical regularities, but relating to a separate object. Here, the statistical regularity manifests itself in a very large number of identical experiments with identical objects, and is in no way relat¬ ed to the interaction of these objects. For example, the electrons in a diifraction experiment may pass through a crystal with arbitrary time intervals and nevertheless give exactly the same statistical picture for the blackening of a photographic plate as if they had passed through the crystal simultaneously. Regularities in alpha disintegration cannot be accounted for by the fact that there are a very large number of nuclei: since there is practi¬ cally no interaction between nuclei inducing the process, the statisti¬ cal character predicted by quantum mechanics is only manifested for a large number of identical objects; it is by no means due to their number. In this connection, a description of phenomena m quantum mechanics involves the concept of probability phase, similar to the concept of the phase of a light wave. In principle, the wave equation can also be applied to systems con¬ sisting of a large number of particles. The solution of such an equation represents a detailed quantum-mechanical description of the state of the system. Let us suppose that as a result of the solution of the wave equation we have obtained a certain spectrum of energy eigen¬ values of the system = (39-1) in states with wave functions •••.'i'". ••• Sec. 39] THE BQUIIJBKI0M DISTBIBTJTION OB MOLECULES 415 Then the wave function for any state, as was shown in Sec. 30, can be represented in the form of a sum of ({/-functions of states with defi¬ nite energy values: ( 39 - 2 ) #1 The square of the modulus W„ = \Cn 1^ (39.3) gives the probability (when the energy of a system in the state t{/ is measured) that the result will be the wth value. The erpansion (39.2) makes it possible to determine not only ampli¬ tudes, but also relative probability phases corresponding to a detafled quantum-mechanical description of the system. The methods of statistical physics make it possible straightway to determine approximately the quantities w„—\cn\^, i.e., the prob¬ abilities themselves omitting their phases. For this reason, the wave function of the system cannot be determined from them, al¬ though it is possible to find the practically important mean values of quantities that characterize macroscopic bodies (for example, their mean energy). In this section we shall consider how to calculate the probability Wn as applied to an ideal gas. Ideal gases. An ideal gas is a system of particles whose interaction can be neglected. The interaction resulting from collisions between molecules is essential only when the statistical distribution Wn is in the process of being established. When this distribution becomes established the effect of interaction is very slight. As regards condensed (i.e., solid and liquid) bodies, the molecules are all the time in vigorous interaction, so that the statistical distri¬ bution depends essentially upon the forces acting between the bodies. But even in a gas the particles must not be regarded as absolutely independent. For example, Pauli’s principle imposes essential limi¬ tations on the possible quantum states of a gas. We shall take these limitations into account when calculating probabilities. The states of separate particles of a gas. In order to distinguish the states of separate particles from the state of the gas as a whole we shall denote their energies by the letter e and the energy of the whole gas by S. Thus, for example, if the gas is contained in a rec¬ tangular potential well (see Sec. 26), then the energy values for each particle are calculated according to equation (26.19). Let e take on the following series of values: e = eo.®i.ea.-•-.efc. (39.4) where there are particles in the state with energy So and in general there are m, particles with energy ek in the gas. Then the total energy of the gas is 416 STATISTICAL PHYSICS [Part IV —^nktk. (39.5) k By giving different combinations of numbers Uk, we will obtain the total energy values forming the series (39.1). We have repeatedly seen that the energy value Ck does not yet define the particle states. For example, the energy of a hydrogen atom depends only upon the principal quantum number n,* so that the atom can have 2n^ states for a given energy [see (33.1)]. This number, 2n*, is called the weight of the state with energy e„. But it is also possible to place the system under such conditions that the energy value defines the energy in principle uniquely. We note, first of all, that in aU atoms except hydrogen the energy depends not only on n, but on I also, i.e., on the azimuthal quantum number. Further, account of the interaction between spin and orbit shows that there is a dependence of the energy upon the total angular- momentum j and, finally, if the atom is placed in an external magnetic field, the energy also depends upon the projection of angular momen¬ tum on the magnetic field. Thus, one energy value mutually corre¬ sponds uniquely to one state of the atom. In a magnetic field all the states with the same principal quantum number are split. We now consider how the states of a gas in a closed vessel are split. We shall suppose that the vessel is of the form of a box with incommensurable squares of the sides af, a|, a\. Then, in accordance with equation (25.19), the energy of the particles is proportional to the quantity —r H—|- -f —~ af a.| a§ where %, « 2 > % ^'’^e positive integers. Any combination of these integers gives one and only one number for the incommensurable values af, a|, a|. Therefore, specification of the energy defines all three integers n^, M 3 . If the particles possess an intrinsic angular mo¬ mentum, we can, so to speak, remove the degeneracy by placing the gas in a magnetic field (an energy eigenvalue to which there cor¬ respond several states of a system is termed degenerate). We shall first consider only completely removed degeneracy. States ol an ideally closed system. We shall now consider the energy spectrum of a gas completely isolated from possible external influences and consisting of absolutely noninteractmg particles. For simplic¬ ity, we shall assume that one value of energy corresponds to each state of the system as a whole, and, conversely, one state corresponds to each energy value. This assumption is true if all the energy eigen¬ values for each particle are incommensmable numbers).* We shall * Not to be confused with nfe! * In a rectangular box the state s (»i, n^, n^) has an energy which is com¬ mensurable with the energy of state e (2wi, 2 n^, 2 n,). Therefore, the energy of all states can be incommensurable only in a box of more complex form than rectangular. Sec. 39] THE EQUILIBRIUM DISTRIBUTION OF MOLECULES 417 call these numbers Sk. Then, if there are nk particles in the i:-th state, the total energy of the gas is equal to *=3 then = 10, which is what we have already seen from direct computation. Particles subject to Pauli exclusion. In the case of particles subject to Pauli exclusion the calculation of is still simpler. Indeed, here we always have the inequality m, ^ gk, because not more than one particle occurs in each state. Therefore, of the total number of gk states Uk are occupied. The number of ways in which we can choose nu states is equal to the combination of gk things nk at a time: 422 STATISTICAL PHYSICS [Part rV gfe! «*!! (gfc—Hfc)! (39.7) There are as many possible different states in the case of nk < ffk, and there is one particle or none at all in any of the ffk states. The most probable distribution of particles among states. The numbers ffk and Uk refer to a single definite energy. The total number of states of a gas is equal to the product of the numbers for all the states separately; P=nPsk'’k- (39.8) le So far we have only used combinatorial analysis. And besides it has been shown that all separate states taken .separately are equally probable. The quantity P depends upon the distribution of particles among the states. It can be seen that, in fact, a gas is always close to a state where the distribution of separate particles among the states corresponds to the maximum value of P ])ossible for a given total energy S’ and for a given total number of particles. We shall explain this statement by a simple example from gambling, as is usually done in probability theory (most easily seen here is the manifestation of large-number laws in a game of chance). Let a coin be tossed N times. The probability that it will fall heads once is equal to 1/2. The probability that it will fall heads all N times is equal to (1/2)^. The probability that it will fall N—1 times heads, and once tails, is equal to (l/2)''^-i X 1/2 x W, because this single occasion can turn out to be anyone, from the first to the last, and the probabilities for mutually exclusive events are additive. The probability for a double tails is equal to “ • The last factor shows how many ways two events can be chosen from the total number N (the number of combinations of N two at a time). In general, the probability that the coin will fall tails k times is |1\N-^‘|1\I‘ Ni 12/ k<.{N-k)C' The sum of all probabilities is, of course, unity: 2^=1 N{N-l) {N-2) 1-2-3 because the sum of binomial coefficients is equal to 2^. Considering the series qk, we can see that ffk increases up to the middle of the sum, i.e., as far as k=NI'2, and then decreases sym¬ metrically with respect to the middle of the sum. Indeed, the A:th Sec. 39] THE EQUILIBRIUM DISTRIBUTION OF MOLECULES 423 N — k+ 1 term is obtained from tbe {k —l)tb term multiplied by-- SO that the terms increase as long as NI2>k. Every separate series for tails appearing is in every way equally probable with all other series. The probability for any given series is equal to (1/2)^. But if we are not interested in the sequence of heads and tails, but only in their total number, then the probabilities will be equal to the numbers qk. For Npl, the function qu has a very sharp maximum at k=NI2 and rapidly falls away on both sides of NI2. If we call the total number of N tosses a “game,” then in the overwhelming majority of games, heads will be obtained ap¬ proximately iV/2 times (if N is large). The probability maximum wiU be sharper, the greater N is. Wo will not, hero, refine this as applied to the game of pitch and toss (see exercise 1), but will return to the calculation of the number of states of a gas. On the basis of the equal probability for the direct and reverse processes between any pair of states, we have sliown that any pre¬ viously defined distribution of particles among states lias the same probability of being established for a given total energy. In the same way, every separate sequence of heads and tails in each separate game is of equal probability. But, if we do not specify the states of a gas by denoting which of the gu states with a given energy are filled, and give only the total number of particles in a state with energy Sfc, then we obtain a probability distribution with a maxi¬ mum similar to the probability distribution of games according to the toted number of occurrences of tails irrespective of their sequence. The only difference is that in the example of pitch and toss the probability depends upon one parameter k, and the probability for the distribution of gas particles among states depends upon all rik. Our problem is to find this distribution for particles with integral and half-integral spins. It is most convenient to look for the maximum of the logarithm of P rather than P itself. In P is a monotonic function of the argument and assumes a maximum value at the same time as the argument P. Stirling’s formula. In calculations we shall require logarithms of factorials. For the factorials of large numbers, there is a con¬ venient approximate formula which we shall deduce here. It is obvious that n In w! = In (1 •2.3*4... w) = k . it-i The logarithms of large numbers vary rather slowly since the difference In(w-l-l) — Inn is inversely proportional to n. Therefore, the sum can be replaced by an integral: 424 STATI8TICAI, PHYSICS [Part IV n n In nl = ^Ink ^ | In kdk = 7i\n n — n = nln ~. (39.9) fc-i b This is the well-known mathematical formula of Stirling in somewhat simplified form. It becomes more accurate the greater n is. Additional maximum conditions. And so we must look for the values of the numbers w* for which the quantity = In P = In n k is a maximum at a given total energy S = ^nkSk k nnd for a given total number of particles N==2Jnk. k This kind of extremal condition is termed bound, because addi¬ tional conditions (39.11) and (39.12) are imposed upon it. Wo shall first of all find rik for particles which are not subject to Pauli exclusion, i.e., those with integral spin. To do this we must substitute the expression for P from (39.C) in (39.10), take the dif¬ ferential dS with respect to all Uk, and equate it to zero. We have (39.10) (39.11) (39.12) iS = In P = In |~] k ({fk + Ilk — 1 )! (fiffe— 1)! Win (gfe + ~ • nfc!(sffc-l)! • *= (39.13) We substitute here the expression for factorials according to Stirling’s formula (39.9): + l)ln -^^ + - ^*~ ^ -rtkhi^ - {gk - l)In-g^ j: ^ j. * (39.14) Since gk is a large number, unity can naturally be neglected every¬ where. We must, of course, differentiate with respect to nk in formula (39.14), because gk is the given number of aU states. Then dS — drik [In {gk -b «*) — In rik] — ^dn*ln —= 0. “ (39.15) It must not be concluded from this equation that the coefficients of every duk are equal to zero, because nk are dependent quantities. Sec. 39] THE EQUILIBRIUM DISTRIBUTION OF MOLECULES 42£ The relationship between them is given by the two equations (39.11] and (39.12) and, in differential form, are as follows: d(^= 2Jekdnk = 0, (39.16; k dN= 2Jdnk = 0. (39.17; k From these equations, we could eliminate any two of the numbers drik, substitute them in (39.15), and afterwards regard the remaining diik as mdependent quantities. Then their coefticients may be regarded as equal to zero. The method ol undotormined coefficients. The elimination of de¬ pendent quantities is most conveniently acliievod by the method of undetermined coefficients. This makes it possible to preserve the symmetry between all 7ik. Let us multiply equation (39.16] by an indefinite coefficient which we denote by 1/0; the meaning of this notation will be cxifiained later. Wo multiply the second equation (39.17) by a coefficient Avhich we denote g/O, so that we have introduced, as is required from the number of supplementary conditions, two quantities, 6 and [l. After this we combine all three equations (39.15)-(39.17) and regard all dnu as independent, and 6 and jx as unknoAvn values which should be determined from equations (39.11) and (39.12). The maximum condition is now written as dS— (39.18] We look for the extremum of one quantity 8 -^ then choose 6 and [x so that the energy and number of particles equal the given values. But if the extremum is determined foi one function without conditions, then all its arguments become mutually independent, and we are entitled to equate any differentia] dtik to zero regardless of the other differentials. Equation (39.18), written in terms of dnk, has the following form: k Bose-Einstein distribution. Let us now put all the differentials except dnk equal to zero. According to what we have just said this is justified. Then, for equation (39.19) to hold, we must put the coefficient of dnk equal to zero: + (39.20) nfc 6 6 ' ' 426 STATISTICAI, PHYSICS [Part TV Naturally, this equation holds for all h. Solving it with respect to rik, we arrive at the required most probable distribution of the number of jiarticles according to state; nfc=—_. (39.21) e » '-1 This formula is called the Boso-Einstein distribution. As to particles for which this distribution is aiiplicable, they are said to obey Bose- Einstcin .statistics or, for short, Bose statistics. They have either integral or zero spin. The unknown quantities 6 and [a, i.o., the para¬ meters in the distribution, are given by equations as functions of N and _Efe?)!_ _ a ^ ‘fe-i* ’ (39.22) ^ ‘k->^ ’ (39.23) * e ® - 1 so that the problem posed of finding the most probable values of Wfe is, in principle, solved. Fermi-Dirac distribution. We shall now find the quantities nu for the case when the particles are subject to Pauli exclasion. In accordance with (39.7) and Stirling’s formula wo have for the quantity 8: * * nil! {gii — > 0 ,)! k - Wfeln-- - {gii - rik) In . (39.24) k Differentiating 8 and substituting into equation (39.18), we obtain JO , gdN "--T+ 0 + (39.25) whence, by the same method, we arrive at the extremum condition: In Ok — nk nk -T + i = 0> and the required distribution appears thus: 9k nk et - H 6 (39.26) + 1 Sec. 39] THE EQUILIBRIUM DISTRIBUTION OF MOLECULES 427 Here, rik^gk as is the case for particles subject to Pauli exclusion. For such particles, formula (29.36) is called a Fermi-Dirac distri¬ bution. The parameters 9 and jjl are determined analogously to (39.22) and (39.23): -= ^ > (39.27) fee « +1 27- = (39.28) fee “ +1 Concerning the parameters 9 and [i. The parameter 9 is an essentially positive quantity, because otherwise it would be impossible to satisfy equations (39.22), (39.23) and (39.27), (39.28). Indeed, there is no upper limit to the energy spectrum of gas particles. For an infinitely large e and 9 < 0, we would obtain e ® = 0, so that, by itself, a Bose di.stribution would lead to the absurd result nu < 0. In (39.23), on the left, we would have the negative infinite quantity ~ which can in no way equal N. Similarly, a Fermi distribution would lead to infinite positive quantities on the left-hand sides of (39.27) and (39.28); and this is impossible for finite N and ^ on the right. Therefore, 6> 0. (39.29) In the following section it will be shown that the quantity 9 is pro¬ portional to the absolute temperature of the gas. The quantity p is very important in the theory of chemical and phase equilibria. These applications will be considered later (see the end of Sec. 46 and the succeeding sections of the book). The weight ol a state. Here we give a few more formulae for the weight of a state of an ideal gas particle. The weight of a state with energy between e and e-pde is given by the formula (25.26), whose left-hand side we shall denote now by dgr (e). In addition we assume that the particles have an eigenmoment j, so that we must take into account the number of possible projections of j, equal to 2/ -f 1: = (39.30) For electrons / = l/2, so that 2/-f-l=2. For light quanta we must use formula (25.24), replacing K in it by w/c and multiplying by two, according to the number of possible polarizations of the quantum: 42S STATISTICAr. PHYSICS [Part IV It is also useful to know the weight of a state whose linear momen¬ tum is between px and px + dpx, and py-\-dpy, 7)^ and pz + dp^. It is determined in accordance with (25.23), also with account taken of the factor 2/-[-l. Thus, for electrons, we obtain dg (p) 9 ydpxdpydj)z (27t;fe)» (39.32) Exercises 1) Write (lewti tlio fortniilu for tlio probability that heads are obtained k times for largo N, where k is close to the maximum ■, where x is a (piantity small com- N pared with . Then, in tho eorrection forms of Stirling’s formula 1/2 In ’2T:k and 1/2 In 2 n {N — k). tho quantity x can bo noglocteil. Wo oxpanil tho ilonom- inator in a series u|) to a:*: N In {N - k )! In hi - x ,n.l = ln(|- + .)l = 4h4-i-.h4-+ J-fllnS.^. Tlio eorrootion terms are 1 / A' \ 1 2 i(ln2,iW-21n2::-i-) = -2ln-^. Substituting this in Iho expression for qk and taking antilogarithms we arrive at the roquiretl formula: 2.x* N q has a maximum at x =-• 0 and dies away on both sides, q is reduced e times in the interval Xe = > characterizing tho sharpness of the maximum. Compared with tho whole interval of variation x, the interval Xe comprises X. 1/2" example, for N = 1,000, the maximum is approximately equal to 1/40. Tho ratio is about 2 so that, basically, tho heads fall between 475 to 52.5 times. Tho probability that heads (or tails) will fall 400 times out ■_ 2 • lo.n no of a thousand is equal to 1/40-e = 1/40 e'®”. In other words, it is Sec. 39] THE EQUILIBRIUM DISTIilBUTIOX OF MOLECULES 429 e"*'"®, i.e., several hundreds of millions of times less than the probability for heads appearing five himdred times. 2) Verify that the probability q has been normalized, i.e., thatj*g (x) dx = \ . Since the probability decreases \ (>ry rapidly with increase in the absolute value of X, the integration can bo extended from — co to oo without notieo- ablo error. Then J q(x)dx = 1/-^ J 6 ^ dx ^ j e- . Wo shall now show that the integral ai>ponring hero is indeed equal to y/ k We shall call it /: /=/«-- Squaring, wo get = j e d 5 j e — j J e (V t V) d^ dti. The integration .sproa- » . (40.1) 'his is the Boltzmann distribution. Let us now determhie the constant from the normalization condition for the distribution: 2^nk=A. (40.2) k Let us sn])j)ose tlmt the gas molecules possess some iiiti'rual degrees f freedom (in addition to the external transport degrees of freedom) lat may be related to electron excitation, the vibration of nuclei ith respect to eacli other, and the rotation of the molecule in s]iace. 'he energy of all these degrees of freedom is quantized. Without efining it more c.xactly for the time being, we can write tlie total aergy of a molecule e in the form of a sum of the energies of trans- dional and mternal motion: Accordingly, the weight of a state of given energy is also n^presented 3 the product of tw'o weights: one relates to translational motion tid is given by the formula (39.32), while the other wo denote simply y go) (we also agree to include in it the factor 2; + l): _F dpx dpy d] X ^ (27 z hf (40.4) Jierefore, formula (40.2) can be written thus: V - -‘^'1 “ 7 “ - - (2,/t)3 g ® J J If N. (40.5) t -fO — oo oo 2 2 2 Expandmg the translational-motion energy into 2m ’ e see that the momentum integral is represented as the product r three integrals of the form 2 j e dpx. — CO 432 STATISTICAL PHYSICS [Part IV These integrals are easily calculated from the second formula of exercise 3, tSec. 39. Eacli of them is equal to V27i:m0, so that con¬ dition (40.5) reduces to the form c a » = V ~N' 6 (40.6) Jf the gas is monatomic then the quantities refer to electron excitations. Therefore, if £0);j> 0, then, actually, only the zero term apjiears in the summation ovi-r the states*. But since the energy is measured from sf") as from zero, the svliole summation, actually, reduces only to tin; zero term y(„). It is of the order of unity. For exanifile, when the ground state has angular momentum 1/2, y(o) = 2. We then olitain the condition for the applicabilitj' of Boltzmann statistics in the form (40.7) For the inc(|uality (40.7) to be satisfied, it is sufficient to satisfy one of two conditions: 1 ) the density of the gas is very small, i.e., the volume ocoiqiied by the gas at a given tcm^ierature 0 is large; 2 ) the temjierature 0 for a given volume V is very high. In the case when the gas is not monatomic, these conditions are _e(i)^ quantitatively ehangctl somewhat because yg{i)e ® is also some i function of 0. But ipialitatively, the conditions of applicability of Boltzmann statistics still hold. Classical and quantum statistics. Wo have seen that for small densities or high temperatures the (piantum distribution laws for a gas ])ass into the classical Boltzmann law. From now on v'e shall agree to call the Bose and Fermi statistics quantum statistics and the Boltzmann statistics, chusical, regardless of wheather the energy spectrum is discrete or continuous. Those statistics will be termed quantum for which the indistinguishability of separate particles is taken into account. In other words, a quantum definition of the state of a system lies at the basis of quantum statistics: the number of jiarticlcs in all quantum states must be given. The classical definition of the state of a system indicates which particles are found in the given states. The lioltzmann formula (40.1) can be obtained from this classical definition. Maxwell distribution. In this section Ave will not be concerned with the statistics of the internal motion of molecules, and Avill consider * The relation between 0 and tenipiTntare is given by formula (40.25). Sec. 40] BOLTZMANN STATISTICS 433 only their translational motion. In accordance with (40.3), the energy of the translational motion of molecules is separable from their internal energy. Therefore, the Boltzmann distribution breaks up into the product of two factors. We are not interested in tlio first of the two factors, but the second, relating to translational motion, is of the form _p]_ g 2 m 0 The weight of a state relating to a given absolute value p is obtained by changing to polar coordinates in formula (39.32); dgiv) (40.8) [cf. (2.5.24)]. Thus, the distribution according to the kinetic energies of transla¬ tional motion is written in the form: p* dn{p) =Ae (40.9) It is applicable both to monatomic and polj’-atomic gases if m is the mass of a molecule as a whole. The constant A is found from the normalization condition A 0 p* _ 2>i'0'dp= (40.10) I'he value of the integral w'as found in exercise 3, Sec. 39. Ji’rom this .ve obtain ■%/ 0 )* (40.11) , ! n place of the momentum distribu- ■ O;! of molecules, it is sometimes useful . ^ave their velocity distribution. For i.ij.’' it is sufficient to substitute p = mv 1 f’e distribution (40.9): dr. iw) = N v^dv. (40.12) Tb'^ distribution had already been deduced by Maxwell, before Bol^uanann, and for this reason it is called the Maxwell distribution. ).u .f’ig. 46 we have plotted the ratio on the ordinate. For KtUu*!! V, this quantity is close to zero becau.se of the factor in the St - 05«0 434 STATISTICAl, PHYSICS [Part I\ equation for the weight of a state; after the zero point it reaches a maximum and exponentially decreases to zero again for large velocities, We thus sec that a gas contains molecules with every possible velocity value. The velocities of gas molecules. The greatest number of molecules have a velocity corresponding to the maximum of the distribution curve shown in Fig. 46. This maximum is determined from equation (40.12). The corresponding velocity is termed the most probable; it is =]/^. (40.13: We find the mean velocity by calculating the integral (we omit the factor N, because the mean value of velocity relates to a single molecule): V = A (40.14; The mean square velocity is also interesting 0 (the result of exercise 3, Sec. 39 is used in the derivation). The ratio V P ; v : tim.p. = V 3 : : V^. The mean energy of a single molecule is equal to and the mean energy for the whole gas is N times greater; ^■= JV£ = -|W0. (40.16; / (40.1^7 This result relates to the energy of tran.slational motion of the mole¬ cules. Numerical evaluations of velocity will be performed belovp. The relationship between energy density and pressure. We tihal now derive a very important relationship between the density o; kinetic energy of a gas and its pressure. This relationship holds for* any statistics and depends only upon the form of the expression for energy in terms of momentum. The pressure of a gas is defined as the force with which the ga.s actf upon unit area perpendicular to its direction. This force is eqval tc the normal (to the surface) component of momentum transmitted by the Sec. 40] BOLTZMANN STATISTICS 435 gas molecules in unit time. Let the direction of the normal to the sur¬ face coincide with the cc-axis. We first choose those molecules which have a velocity component along the a:-axis equal to Vx. They will reach the surface of a volume in unit time if they initially were situated in a layer of width Vx- Let us cut out a cylinder from this layer with base of miit area and height equal to Vx- The volume of this cylinder is Vx- If dn. (u*) is now the number of molecules whose velocity compo¬ nent normal to the surface is then the density of these molecules is There are Vx such molecules in a cylinder of volume Vx. Each of them, upon elastically colliding with a wall, will reverse its normal velocity component, and the wall will receive a momentum mvx — (— mvx) = 2mvx- (40.18) Thus, all the gas molecules having a velocity Vx, transfer to the wall in unit time a momentum „ dn{vx) c\ 2 dn(vx) 2mvx — ^—^•Vx=2mv% — Y~ • (40.19) In order to obtain the gas pressure on the wall we must integrate (40.19) over all Vx from 0 to oo, and not from — oo to oo, because molecules moving away from the wall will not strike it. Thus, the pres¬ sure of the gas on the wall is oo oo p = -^jv%dn(vx) = ~jvldn{vx). (40.20) 0 —oo On the other hand, the mean kinetic energy of the gas is oo oo ' oo ^ = ~ j vldn (vx) + -^ J Uydn (Wy) + -^ j vldn (vx) = = \ v%dn{vx), (40.21) —oo because the mean values of the squares of all the velocity components are identical. Comparing now (40.20) and (40.21) we find that the gas pressure is equal to two thirds of the density of its kinetic, energy: 2 S’ P = Yir- (40.22) This result was published by D. Bernoulli, as early as 1738, a century and a half before statistical physics began to develop as an independ¬ ent science. 28 * 436 STATISTIOAI, PHYSICS [Part IV Only two assumptions have been used in the derivation of (40.22): identical values of the three velocity projections are equiprobable and the kinetic energy is equal to . The concrete form of the distribution function is not essential. The Clapeyron equation. If a gas is subject to Boltzmann statistics, then, in accordance with (40.17), the mean kinetic energy i is equal to —^ . Substituting this in (40.22) we obtain pF = iVe. (40.23) But from the definition of absolute temperature pF = RT. (40.24) From this wo obtain the relationship between “statistical” temper¬ ature 0, measured in ergs, and the temperature T, measured in degrees Centigrade: 6 = 4^ = -6^4 x'io- = T . (40.25) R The ratio I;=-^-is called Boltzmann’s constant. It is equal to 1.38 xlO-i*. The temperature can also be measured in electron-volts, one electron-volt being equal to 1.59 x 10“^^ erg. Translating ergs into degrees with the aid of Boltzmann’s constant, we find that 1 ev = = 11,600°. As is known, the specific heat of an ideal monatomic gas is equal 3 3 to -g- R, thus corresponding to an energy ^ RT. Replacing RT by ATO, we find ^ = -|-A70 in agreement with (40.17). The relationship (40.25) allows us to calculate the mean velocity of gas molecules without using the Avogadro number N. Indeed, M — ]I^RT ^ 1 Ttm [ 7cA?m \ kM ’ where M is the molecular weight of the gas. For example, the mean velocity of hydrogen molecules at a temperature of 300° K is - 1/8-¥.3-10’“-300 , , V = -2 -- = l,800w/sec. This value is comparable with the exit velocity of a gas into a vacuum or with the velocity of sound [see (47.30)]. The thermonuclear reaction. When nuclei collide reactions are pos¬ sible between them that proceed with the release of energy. For exam¬ ple, in a douteron-deuteron collision one of two reactions can occur (besides elastic scattering): Sec. 40] BOLTZMANN STATISTICS 437 Df+Df = ) He’ + no, ^H! +ffl. Here 11\ is tritium and nj a neutron. Another example is Li§ + D! = 2HeJ. In order that charged nuclei may be able effectively to collide, they must overcome the potential barrier of Coulomb repulsion, which was considered in Sec. 28. The dependence upon energy for the probability of passing through the potential barrier is basically determined by the barrier factor [sec the first term on the right hi (28.12)]; _ 2k ZiZ, (* e . (40.26) Here, Z-^e and Z^e are the charges of the colliding nuclei and V|| is the relative velocity along their joining line [recall that (28.12) refers to one-dimensionnl motion]. The reaction can be produced by accelerating the particles in a dis¬ charge tube. But charged ])articles, striking a substance, mainly spend their energy on ionization and excitation of the atoms. And since, according to (40.26), the jirobability of the reaction at small energy is vanishingly small, the majority of incident particles do not cause a reaction. Of the total number of particles it turns out that 10 ~®—10'® are effective. Therefore, the energy yield of the reaction is considerably less than the total energy spent in accelerating the beam of particles. The situation is different if the substance used for the reaction is at a very high temperature, of the order of 10’ degrees. At this tem¬ perature, the nuclei of the heated substance already react at a suffi¬ ciently high rate, and transmission of energy to electrons does not occur because their mean energy is the same as that of the nuclei. Let us calculate the rate of a nuclear reaction occurring under such conditions. It is termed thermonuclPMr. Let the effective cross-section for the reaction between nuclei with relative velocity ?;j| be a (wn). We assume that different nuclei react: we shall call them 1 and 2. Let us construct on each nucleus 2 a cyhnder with base area a (i^u) and height numerically equal to ^n. Then, by definition of a (i;||), all those nuclei 1 which occur in the volume of these cylinders and which have velocity V\i relative to nuclei 2 will be involved in the reaction in unit time. The number of such events in unit volume and unit time is equal to the product of Wi • V|| (T (U||) dq (i?,,), (40.27) where and are the numbers of nuclei 1 and 2 in unit volume, and dq (Vji) is the probability that the relative velocity is equal to V\\. 438 STATISTICS AT PHYSICS [Part IV Indeed, a cylinder of volume V\\ a (V||) can be constructed on each nucleus 2, and there will be a (^n) nuclei 1 in each cylinder. The velocity distribution of the nuclei is taken into account by multiplying by dq (tsu). If 1 and 2 are identical nuclei, then expression (40.27) must be halved so that each reaction is not taken into account twice. We indicate this by the factor (2) in the denominator of expression (40.28). Let us now determine the probability factor dq (iS||). The absolute- velocity distribution is given by the product of two Maxwellian fac¬ tors of the form m,vl “•■'I In the exponent of this expression is the sum of the energies of both nuclei. In accordance with formula (3.17), it can be split into the kine¬ tic energy of the motion of the centre of mass of the nuclei and the kinetic energy of their relative motion. Hence, in the product a factor is separated that gives the relative-velocity distribution: mi'« ""'i g 2(m,+ m,)6^g 26 ^ ^ 26 ^ 26 ^ where m = [see (3.20)], v^^v\ + vl. Normahzing the distribution over V\\ to unity and passing to the re¬ duced mass m, we obtain an expression for the probability that the value of relative velocity along the line joining the nuclei will be U||. dq (t’li) = e dv||. The barrier factor (40.26) depends upon V[\. Thus, the overall rate of a thermonuclear reaction is oo 0 Taking into account the barrier factor, we write the dependence of effective cross-section upon the rate as 2nZi2,e» The factor Uq here depends considerably less upon the rate than the exponential function. Sec. 40] BOLTZMANN STATISTICS 439 The integral in (40.28) now reduces to the form; T — Oo(vil)i;iie hv\ 20 di?||. (40.29) 0 It can be calculated, to a good approximation in the case when the temperature is so low, that the greater part of tlie reaction jirooeeds at the “tad” of the Maxwelhan distribution at a rate greater than the mean. Let us show how this calculation is done. We denote tlie argument of the exponential in the integrand thus; / («il) 2-!tZiZ2e^ _ a , ft 2 _ 2 TcZ^Z^e^ , w = ^ ® = T • We find the minimum of the function / (wn) from the condition |(-=-4 + H, = 0; .!=(/»:. (40.30) We shall see that the basie contribution to the integral is given by values of v\\ close to wjj. Near the minimum, / (wn) can be represented in the form / (t’li) = / (*’1!) + Y (^11 - (-£i-)o = =-|-Va*fe + -1-6(1)11 —i;jl)2, (40.31) and the integral (40.29) is written as r —/(■'ji)—I'll)* J cfo(«ii)«iie di;||. (40.32) 0 The minimum of / (wu) corresponds to the rate v\ at which the greatest number of reactions occur. The ratio of the rate wlj to the mean relative rate | V\\ \ is, according to (40.30), ji - ]/-J VaVh ~ (f)''-(f), (40.33) smee ■'”•-1/11'= 1/1- We shall call the temperature low if the ratio 0 n i«ii I is several times greater than unity. Then the maximum of the integrand of (40.32) 440 STATISTICAL PHYSICS [Part IV is very sharp at the point V\\=v\, because it decreases e times when W|l deviates from vH by an amount j/ , which is considerably less than W|). It was therefore justifiable to terminate the expansion in (40.31) with the second term. In addition, the quantities og (vn) and V\\ can be taken outside the integral sign when U|| = t)il. The error in both approximations is of the order —. The integration can be taken V|| from — oo to CO because the integrand rapidly decreases as wn re¬ cedes from v“, so that the error is exponentially small. Thus, -1(4- / 0 ^ (*^ 11 ) *’11 0 d«li: Zh ■'ll’ dO|i = -/(ril) (40.34) Substituting the values a and b and using (40.28), we find the expression for the rate of a thermonuclear reaction _WiW2 (2) Vs ® V( e* y ' 5 ! V h ~ ) e . n > *’il = hm (40.35) The exponential factor depends very strongly upon the temperature. For example, for a reaction in deuterium, this factor clianges 3,600 times when the temperature is increased from 100 to 200 ev. Thermonuclear reactions are the source of stellar energy and for this reason play as important a part in our life as chemical reactions! Ideal gas in an external field. We shall now consider an ideal gas acted upon by an external field with potential U. The potential ener¬ gy can depend both upon the position of the molecules in space as well as their orientation (if the gas is not monatomic). The total energy of a molecule is e = -£- + e<--) + C/ . (40.36) If U depends upon the position of a molecule in space, i.e., U=U (x, y, z), then we must pass from a finite volume V, m the weight factor (40.4), to an infinitely small volume dV=dxdy dz. Then part of the distribution function that depends upon coordinates x, y, z can be Sec. 40] BOLTZMANN STATISTICS 441 separated, and a formula is obtained defining the dependence of gas density upon coordinates: dn{x,y,z) = 71^6 ® dxdydz. (40.37) Here, the potential energy calibration is U (0, 0, 0) = 0, and tlie gas density at this point is equal to Wq. For example, in a gravitational field, U=mgz, so that mjjz dn{z) — 7if^e ** dz. (40.38) It should be noted that in the earth’s atmosphere the “barometric” formula (40.38) is rather more applicable qualitatively because air temperature is not constant with height. In addition, the “barometric” formula indicates that the composition of the air must vary with height as a result of the different molecular weights of nitrogen, oxygen, and other gases. Actually, the air com¬ position is almost uniform vertically because of vigourous mixing processes. The nonequilibrium state of planetary atmospheres. In place of the approximate ex]iression for the i)otential energy in a gravitational field, let us substitute its exact expression (3.4). Let us first of all express the constant a in formula (3.4) in terms of more convenient quantities. The force of gravity at the earth’s surface is —mg and, from the general gravitational law, it is equal to —, where is the radius of the earth. From this a=xmgr\,so that 17= — —Therefore, the gas density must vary with height according to the law n — n^e^'^. (40.39) This quantity remains finite even at an infinite distance from the earth, and since the exponent is equal to unity at infinity we have called the proportionality factor Woo . Near the earth, where r = r^, the density is greater than ?ioo by as many times as the quantity Hi^fo Mgrp e ® is greater than unity. The radius of the earth m 6.4 x 10® cm, g <=» 10® cm/sec®. From this we obtain for oxygen Mgrl 32 • 10“ • 6.4 • 10* RT ■ — 8.3.10’ • 300 • 442 STATISTICAli PHYSICS [Part IV In actual fact the density of the earth’s atmosphere at infinity is equal to zero. Therefore, it follows from formula (40.39) that the atmosi>here cannot arrive at the most probable state when i?! the earth’s gravitational field, and is gradually dis])ersed into space. The most probable state of a gas is called statistical equilibrium (see 8 ec. 45). The equilibrium density of the atmosphere at infinity is e®®® times less than at the earth’s surface. Therefore the present state of the atmosphere is very close to equilibrium. For the moon, equilibrium has been reached: its atmosphere has comifietely escaped! A kinetic interpretation of the dispersion of planetary atmospheres. It is easy to understand the reason for the recession of gases to infinity. Any particle who.se velocity exceeds 11.5 km/sec is capable of over¬ coming the earth’s attraction: its motion is infinite. In accordance with the Maxwell distribution (40.12) a gas will always have molecules with every possible velocity. In literal notation, the velocity of mole¬ cules capable of going to infinity is defined by the equation (40.40) Taking u® from this equation and substituting into the Maxwell distribution, we once again obtain the exponential e ® for the fraction of molecules capable of leaving the atmosjihere. It is easy to estimate the number of such molecules in the atmosphere at any instant of time. The earth’s surface is 5x10^® cm®. There is about 1,030 gm of air above every square centimetre, i.e., about 35 moles. Hence, the total number of molecules in the atmosphere is 5 x 10^® x X35 x 6 X10®® = 10**, and the fraction of molecules of velocity greater than 11.5 km/sec is e ®®®=10“®**. Therefore the mean number of mole¬ cules capable of leaving the earth at each instant is only 10“®®®. Of course, those molecules close to the earth’s surface will not be able to “carry” their energy to the upper layers of the atmosphere because of coUisions with other molecules. The dielectric constant of a gas. We shall now consider a gas whose molecules have a constant dipole moment in a constant and uniform electric field. Those molecules can have characteristic dipole moments for which there is some preferred direction: NO, CO, HjO (along the altitude of the triangle passing through O), NHg (along the axis of symmetry of a three-sided pyramid). The more symmetrical mole¬ cules do not possess moments: Hg, Oj, CHj (tetrahedron), COj (this proves that the COj molecule has the form of a rod with the C atom in the centre). Rotational motion is quantized. In the next section it will be shown that for all gases except hydrogen, at a temperature of several tens of degrees (from absolute zero), the states with large quantum numbers Seo. 40] BOLTZMANN STATISTICS 443 are already excited. In these states the motion may be rega ded as classical. Then the total rotational energy of a molecule simply breaks down into the kinetic energy of rotation (see Sec. 9) and potential energy, which depends upon the orientation of the dipole moment relative to the external electric field: U— — (dE) = — d’Fj cos & [see (14.28)]. In classical motion, the potential and kinetic energies may be regarded as quantities instead of operators. Therefore, in the Boltzmann distribution the factor that depends only upon potential energy is split off: d • E cos S' dn(%^)~Ae ® sin^-d-S-. (40.41) Here, sin d is proportional to the element of solid angle in which the vector d lies [cf. (6.15)]. Let us now determine the electric polarization of a gas in an external field. For this we must calculate the mean projection of the dipole moment upon the electric field, i.e., d ■ j e ® cos a sin 0 d a d • cosO^ = —5---, (40.42) Je ® sinada 0 It turns out that it is sufficient merely to find an expression for the integral in the denominator, because equation (40.42) can be rewritten thus: d-cos&-- e~ln dE d* B cos^ ' fe ® sinO'dO' 0 (40.43) Indeed, differentiating the integral with respect to the parameter E, we revert to (40.42). The integral can be calculated using the fact that sin d9-== —d (cos ff): ” d’Ecos^ sin (/•Ecosa’^ Ed 20 . . Ed Ed^^-r (40.44) The integral in formula (40.43) is called a statistical integral. For quantized energy levels, it is replaced in the general case by a statisti¬ cal sum. The expression for the summation and integral will be met with many times again. It is very convenient in calculating mean values. 444 STATISTICAL PHYSICS [Part IV Substituting (40.44) in (40.43), we obtain an expression for the mean projection of the dipole moment on the electric field d-cosO'= d • |coth^ —. (40.45) This expression was obtained by Langevin. Let us investigate the right-hand side in two limiting cases: E<^ -^(weak field) and -j (strong field). If the field is weak then we can use tlie expansion of coth x in terms of x: cothx = ^ + ^, whence d^c^ = ~ . (40.46) The polarization of the gas is P == iVd-cosO- = , (40.47) and the dielectric constant is calculated from the definitions of induc¬ tion (16.23) and (16.29); P = 47tP = p(l+^^^) = eP. (40.48) In a strong field coth tends to unity and tends to zero. Therefore, cos 0- tends to unity. This means that aU the dipoles are orientated along the external field and saturation sets in. Then D=E + 4:n Nd. We notice that, for P = 300°K, this case would correspond to a field P > 10’ v/cm, which is considerably greater than the breakdown potential. Paramagnetism of gases. We shall now find how the magnetic per¬ meability X is calculated. Here we must take into account the fact that the magnetic moment is related to the mechanical moment of electrons, and the latter is a quantized quantity, i.e., it takes on a discrete series of values. Usually an electronic mechanical moment does not have a value greater than several units, so that the limiting transition to classical theory cannot be performed. An atom can also have a magnetic moment (as opposed to an electric dipole moment). There¬ fore, let us determine the magnetic susceptibility arising from the orientation of atomic magnetic moments in an external field H. Let us suppose that an atom in the ground state possesses an orbi¬ tal angular momentum L, a spin angular momentum S and a total angular momentum J. In other words, the ground state is a multiplet state. Let the multiplet splitting (fine structure) be defined by the Sec. 40] BOLTZMANN STATISTICS 445 energy interval A, so that the level with the closest value ^ ± 1 differs from the ground level by the quantity A. If the energy of the ground level is e^, then the closest level has an energy Eq + A. The ratio of the number of atoms in the ground state to the number of atoms in the closest state, belonging to the given multiplet, is, according to (40.1) 2J+1 e ® _ 2J+1 -!■ 2(J±l) + l' (ep + A) ~ 2(J+ 1) + 1 ^ e 8 (40.49) Thus, if the multiplet splitting A is considerably greater than 6, the majority of the atoms are in the ground state. K they are placed in an external magnetic field then each of the multiplet levels is split into 2 J +1 levels, corresponding to its value of J. Suppose that the field corresponds to the anomalous Zeeman effect in the sense that the splitting of each multiplet level in the magnetic field is con¬ siderably less than the fine-structure splitting A as defined in Sec. 35. Then, from (35.11), the energy of an atom in the ground state is ^ = (40.50) where gL is the Lande factor [see (36.12)] and po is the Bohr magneton. The number of such atoms is given by the Boltzmann distribution n {Jx) = Ae 9 . (40.51) We must again determine the mean value of the magnetic moment projection on the field: J 1^0 —J We have put the minus sign on the left because the electronic charge is negative. Formula (40.62) involves a statistical sum. The summation is performed only over those levels which are obtained when the ground state of the multiplet is split in the magnetic field, since the number of atoms in an excited state is small. Summing the geometric progression, it is not difficult here to obtain a general formula similar to the Langevin formula (40.45). But we shall confine ourselves to the case of a weak field, when the exponen¬ tial function is expanded in a series. The expansion must be taken 446 STATISTICAL PHYSICS [Part IV up to the second term inclusive because the sum of the terms which are linear in is equal to zero: —j V^9lHJz , 1 / v.(,gLB:Jz\i~\ 'z , z \^~\ _ 2 V 9 / J 2«/ -f" 1 “h {MlHY 20 >= J We calculated the sum of Jz^ in Sec. 30 [see (30.27)]. Using the value for the sum then obtained, we write the required mean moment thus: — gi, = 6 In ^2 J + 1 + 1 ( 2 /+ 1 ) 6 0 * _ 1 i,lglnj{j+l) i 0 (40.63) where we have once again neglected terms of higher order in H. Formula (40.63) is completely analogous to (40.46) for the electric moment of dipole molecules produced by a field. The characteristic magnetic moment is represented by the quantity iiqQlvJ {J + 1) so that the Lande faetor gt takes into aecomit the spin magnetic anomaly. Thus, magnetic susceptibility can be calculated from data obtained from spectroscopic observations. Paramagnetism oi rare earths. There are almost no elements for which we can completely verify formula (40.63) as applied to the gaseous state. But in rare earths the moment of the electronic cloud is due to the 4 /-shell, which occurs, as mentioned in Sec. 33, deep inside the atom. When such an atom is part of a crystal lattice, the 4 /-shell is but slightly subjected to the action of the electric field of the neighbouring atoms so that its state may be regarded as being almost the same as for a free atom of a rare-earth element. Thei’cfore, (40.53) is applicable to those chemical compounds of rare-earth ele¬ ments where other elements do not possess a characteristic magnetic moment. Its agreement with ex])eriment is very satisfactory for almost all the elements of the rare-earth group. Exercises 1) Find tlie mean relative velocity of two molecules of dilTorent gases occur¬ ring in a mixture. The relative velocity distribution is given by a formula similar to the I’n distribution but written for all three velocity components. This formula is similar to (40.12), but it involves the reduced mass w = ——instead nil + ni^ of the mass of a single molecule. Hence, like (40.14), the mean relative velocity tmms out equal to Sec. 41] BOLTZMANN STATISTICS 447 86 Ttm 1/8 9 (m, + Wg) y re »ii Wj If the molecules are identical, their mean relative velocity is ^2 times the moan absolute velocity. 2) Calculate the velocity of a bimolecular reaction r', if the effective crosa- section depends upon the velocity component (along the lino joining the nuclei) in the following way: (Vii) 0 Vn < do Vi, > r m 1 ^'. Then, from the general formula (40.28), we find The decisive quantity in this result is the exponential factor e ® . The quantity A is called the activation energy. It is equ^ to the height of tho poten¬ tial barrier over which the colliding pai'ticles must pass in order that tho reaction may occiu'. Unlike a thermonuclear reaction, it is assumed here that the motion 3f the reacting particles is classical. Transitions below the barrier make a vanish¬ ingly small contribution in chemical reactions. See. 41. Boltzmann Statistics (Vibrational and Rotational Molecular Motion) Molecular energy levels. In order to apply statistics to gases consist¬ ing of molecules, we must classify the energy levels of the molecules. The fact that a nucleus is considerably heavier than an electron, and, therefore, moves much slower, is very helpful here. We have used this in Sec. 33, when considering the question of the binding energy of two hydrogen atoms in a hydrogen molecule. The eigenfunction can be found for any relative positions of the nuclei. In a diatomic molecule the position of the nuclei is defined by a single parameter— the distance between them. The energy eigenvalue of the electrons depends upon this distance. Adding the energy of Coulomb repulsion of the nuclei to the electron energy, we obtain, for a given electron wave- function, the energy of the molecule as a function of the distance be¬ tween the nuclei. For example, in a hydrogen molecule, the curves for this relationship are of different form in the case of parallel and anti- parallel spin orientations (Fig. 47). The lower curve refers to the state with a symmetric spatial wave function and antiparallel spins, while the upper curve relates to the states with an antisymmetric spatial 448 STATISTIOAI, PHYSICS [Part IV function and parallel spins. The lower curve has a minimum at r = r„ so that hydrogen atoms may form a molecule only in a definite elec¬ tron state. In the general case, several different electronic states can have a minimum. The distances between corresponding potentional curves are defined from a wave equation of the type (33.23). In this equation we can neglect terms involving the masses of the nuclei in the denomi¬ nator. Therefore, the energy scale separating different electronic states of the molecules is the same as for an atom, i.e., from one to ten elec¬ tron-volts. Close to the miniraum of potential energy, nuclei may perform small oscillations. To a first approximation, these oscillations are harmonic so that their energy is given by the general formula (26.21): hvi{v (41.1) Here, v is called the vibrational quantum number of the molecule. This number is, naturally, integral. Fig. 47 shows a more general dependence of energy upon v, taking into account that the potential energy curve is not a parabola as in Fig. 41. However, practically, the deviations from formula (41.1) affect but little the statistical ^ 1 quantities, because dissociation occurs when oscillations with 11 large v are excited (see Sec. 51). 1\ The frequency w depends upon the electronic state in l\ which nuclear oscillations occur. In accordance with the \\ general formulae for frequency (7.10)-(7.12), \ the fre- Fig. 47 quency is inversely proportional to the square root of the reduced mass of the nuclei. There¬ fore, the vibrational quantum is considerably less than the distance between electronic lev¬ els, which is independent of the nuclear mass. It is of the order of tenths of an electron-volt. In addition to vibrational motion, a molecule with two atoms may also perform rotational motion, notation is most simply taken into account when the resultant spin of the electrons is equal to zero and the total orbital angular-momentum projection of the electrons on a line joining the nuclei is also equal to zero. These conditions are satis¬ fied in the electronic ground state for nearly all molecules, with the exception of Og, the resultant spin of which is equal to unity (but the projection of the electronic moment on the axis is zero), and NO, where the spin is one half (and the orbital angular-momentum pro¬ jection of the electrons on the axis is zero). Disregarding these excep¬ tions, we may consider a molecule of two atoms as a solid rotator, i.e., a system of two point masses at a fixed distance re corresponding iSoc. 41] BOLTZMANN STATISTICS 449 to the mimmum of the lower potential curve in Fig. 47 (see exercise 2, Sec. 30); (our case corresponds to J 3 = 0 and 1;=0, so that the closest excited level in k with A: = 1 is moved to infinity. The rotational moment of the rotator is perpendicular to the line joining the nuclei since its projection on this line is equal to zero). As we know from Sec. 5 [see (5.6)] the rotational energy of two particles is _ where m is the redueed mass and mr\ is equal to the moment of inertia of the rotator Going over to the quantum formula, we substitute the angular-momentum eigenvalue. It is usual to denote it by the letter K, so that 2 jrerl (41.2) This formula corresponds to the energy of a symmetric top with A -=0 (cf. exercise 2, Sec. 30). It mvolves the mass of the nuclei in the denominator. Therefore, the distances between neighbouring rotation¬ al levels are of the order of a thousandth of an electron-volt and less. Thus, to a good approximation, the total energy of a two-atom molecule can be written in the form of a sum with three terms: S — \- £r h^K (K+ I) ' 2 m r'} (41.3) where £c (be., it is independent of the nuclear mass m), m The excitation of electronic levels. If we substitute the expression (41.3) in the Boltzmann distribution, the latter separates into the prod¬ uct of thi’ee distributions involving electronic, rotational, and vibra¬ tional states. Let us suppose that a gas is considered with temperature not exceeding several thousand degrees, for example, 2,000-3,000°. Then if the energy of electronic excitation is several electron-volts (1 ev =11,600°, since the temperature can be defined in energy units), the fraction of molecules in excited electronic states is a very small ze number: e ® . In those cases when there are very low electron levels, the Boltzmann factor may also be other than a small quantity. But, as a rule, dissociation of the molecules sets in earlier than excitation of their electronic levels (see Sec. 51). Excitation of vibrational levels. Let us examine the vibrational states. For generality we may consider not only molecules ivith two atoms, but also polyatomic molecules. If the oscillations of such molecules are harmonic, we can make the transition to normal coordinates, as was shown in Sec. 7. Then the vibrational energy assumes the form of 29 - 0060 450 STATISTICAI, PHYSICS [Part IV a sum of the energies of independent harmonic oscillators. The energy levels for each such harmonic oscillator are given by a formula of the form (41.1) with a frequency w correspondmg to a given normal oscilla¬ tion. Molecular oscillations are basically divided into two types: “valent,” in which the distances between neighbouring nuclei mutually change, and “deformational,” where only the angles between the “valence directions” change. For example, in a COg molecule, having a straight- line equilibrium form 0 = 0 = 0, valent oscillations alter the distance between the carbon nucleus and the oxygen nuclei, while deformation¬ al oscillations move the € nucleus out of the straight-line configura¬ tion. The frequencies of deformational vibrations are several times less than those of valent oscillations. The estimation /^co~0.1ev related to valent oscillations. In any case, if the vibrational energy breaks up into the sum of energies of separate independent oscillations, then the distribution function also splits into the product of distribution functions for each separate oscillation. Let us calculate the mean energy for one normal oscillation: oo '■“(■'+1) - . - CO '>“(■'+t) * ..-0 here we have used the same transformation as in the derivation of (40.43) and (40.62). Formula (41.4) involves the statistical sum for a harmonic oscillator. The sum of the geometric progression inside the logarithm sign is very easily calculated. Indeed, /l do (41.4) t /-0 t '^0 1-e 0 Substituting this in (41.4) and differentiating, we get tv ha ha (41.5) (41.6) The first term in (41.6) simply denotes the zero energy of an oscilla¬ tion of given frequency. The oscillation possesses this energy at abso¬ lute zero because then the second term in (41.6) does not contribute Sec. 41] BOLTZMAim STATISTICS 461 anything. The second term has a very simple meaning. If we write the mean energy in terms of the mean vibrational quantum number 'v then it is obvious that — ft a . — eK= ^~ + h CO e„ I dp I dff sioC ® , , ] dp] die- «■']. ( 41 . 11 ) J dp J dqe ® CO CO — oo -• oo The statistical integral inside the logarithm is calculated in the usual way: 25 * 452 STATISTICAL PHYSICS i [Part IV ” __r ” ._-I/ 9 ft- 9 J e je dq — V^v:mQ.y—^=-^Q. (41.12) - OO — fjO Whence eo> =6. Then the total vibrational energy of a gas occurring at a frequency w is '^~^=-.NQ = lcT, (41.13) and its contribution to the specific heat is corresx)ondingly equal to B [see (40.17)]. Thus, at a high temperature the specific heat due to vibrational degrees of freedom tends to a constant limit. The excitation of rotational levels.* Let us now consider rotational energy. The weight of a state with a given value of moment K is, as usual, equal to 2 /v +1, in accordance with the number of jiossiblo projections of K. Especially interesting is the case when a diatomic molecule consists of two identical nuclei. In classif 3 dng the states of such a molecule it is necessary to take nuclear spin into account. Indeed, the wave equation for a molecule consisting of identical atoms does not change form when the nuclei are interchanged. Therefore, if the nuclei have half-integral spin, the wave function must be anti¬ symmetric with respect to the interchange of both nuclei, while if they have integral spin it must be symmetric. The symmetry of the eigenfunction of a molecule is determined by the symmetry of its factors fin the approximation (41.3) it is separated into factors]: electronic, vibrational, rotational, and nuclear sjiin. The electronic term of most molecules docs not change when the nuclei are inter¬ changed. The vibrational function depends only uiion the absolute value of the distance between the nuclei and therefore does not change either. The rotational eigenfunction is even with respect to this permutation in the case of even K, and odd for odd K. Therefore, if the nuclear spin is half-integral, then the spin function must be anti¬ symmetric for even K and symmetric for odd K, so that the resultant wave function may always be antisymmetric. If the nuclear spin is integral, the position is reversed, and if it is equal to zero, then odd K are in general excluded because then the spin factor simply does not exist. Rotational energy of para- and ortho-hydrogen. We shall now consider the rotational states of a hydrogen molecule. The total nuclear spin for hydrogen can equal unity (the ortho-state) and zero (the jiara- state). The weight of a state with spin 1 is 3 and that with spin 0 is 1. The state with if = 0 is even in the rotational wave function. Hence, it must be odd in the sjiin function, i.e., it must have spin 0, (see Sec. 33). But the state with zero moment possesses the least rotational * The hypothesis that the rotation of molecules participates in the thermal motion of a gas was put forward by M. V. Lomonosov as far back as 1745. Sec. 41] BOLTZMANN STATISTICS 453 energy. Therefore, only para-hydrogen is stable close to absolute zero. At a temperature other than zero all those states, for which the _ h' K (K + 1) Boltzmann factor e is of tlie order unity, are excited. Taking the moment of inertia of a hydrogen molecule to bo equal to 0.45 x >10-^0, we can see that already at 2’ —300° K the summation over all odd moments 2^(2K+l)e^ i:=i, 3, differs from the summation over even moments by several thousandths. But since the states with even moments are, for hydrogen, nuclear- spin ortho-states, each state with even moment has an additional weight factor 3 according to the number of projections of spin 1. Thus, at room temperature, 3/4 of hydrogen is ortho-hydrogen and 1/4 para-hydrogen. If hydrogen is rapidly cooled the ratio 3:1 is retained for a long time because the ortho-para-transition proceeds slowly. Such a state is obviously nonequilibrium since all the hydro¬ gen m an equilibrium state, at a temperature close to absolute zero, must be in the para-state. One of the methods of obtaining pure para-hydrogen is to adsorb hydrogen onto any substance that disrupts the molecular bonds during adsorption, for example, activated carbon. When desorbing the hydrogen by pressure reduction at low temperature, the change is that to the para-state. If the hydrogen is then heated to room temper¬ ature it stays in the para-state for quite a long time. Let us now write down the formulae for the mean rotational energy of ortho- and para-hydrogen. For simplicity we shall denote the factor in the rotational energy by the letter B. Then 2J(2K+l)e * -liKiK + l) K~0,2,i. (2A+l)e X^O.2,4, .. -fC(K + l) = 02 e 80 In r (2A+ 1) -f k:(k:+1) e *’ (41.14) The difference between e ortho and e para is that the summation is performed over odd K. For a mixture at room temperature I _ 3. Sr- - Spara r Sorttio- (41.16) 464 STATISTICAI. PHYSICS [Part IV At very low temperature, it is sufficient to retain only the term with A^ = 2 in the summation (41.14), so that f _ ® j 5 .\ ■spara- 0* In 11 + 5e » 1 30£e » . (41.16) For ortho-hydrogen we obtain / _2C iortho=0*^ln(3e «+le « J ^ W Vill ^ - ®--r^2il(l -f 14 e 3e~ ® The determination of nuclear spins from rotational specific heat. The rotational specific heat of hydrogen makes it possible to determine the spin of a ])rotoii. Let ns consider formula (41.17). In it, the first term is a constant. It is due to the fact that a molecule of ortho¬ hydrogen would have a rotational energy 2 J3 even at absolute zero. This energy does not contribute to the specific heat because it does not depend upon the temperature. Defining specific heat as the deriv'- ative ^, we see that for a sufficiently low temperature the ratio of the specific heats of ortho- to para-hydrogen tends to zero as _4B e Therefore, if ordinary hydrogen is rapidly cooled to a low temperature, its rotational specific heat will be determined by a quarter of its mole¬ cules in the para-state. It will be four times less than the rotational specific heat of pure para-hydrogen at the same temperatiire. Thus, by measuring the specific heat of the equilibrium state of hydrogen at low temperature (i.e., the para-state) and of rapidly cooled hydrogen, we can determine the spin of a proton or, knowing the spin from other data, we can show that protons are subject to Pauli exclusion because they possess an antisymmetric wave function. The rotational specific heat of molecules consisting of different atoms. Diatomic molecules that do not consist of identical atoms possess equal nuclear-spin weights for states with odd and even K. Therefore, their mean rotational energy is expressed thus; °° BK(K+l)-j (2/i:-fl)e » . (41.18) - K = 0 J The sum inside the logarithm cannot be written in finite form, but it is easily tabulated. Let us evaluate the temperature at which use 10B\ (41.17) Seo. 41] BOLTZMANN STATISTICS 455 of an integral as a substitute for the summation is justified. Thus, for hydrogen 2TOr? = 1.67-10-“(0.74)“-10-i« ‘ which corresponds to a temperature of 87° K. Here, m is the reduced mass of two protons, equal to half the proton mass; r* '~0.74x 10-® cm (where we obtained the moment of inertia used above). For other gases B is of the order of several degrees so that for all temperatures at which these gases are not in the liquid state the ratio JB/6 is a small quantity. To a good approximation, the summation in (41.18) may be replaced by an integral. If we take then and K [K + 1) = .r, (2 + 1) clK -= 2K + 1 = rfx (dK — 1) 27(2/v + 1). BK(lCf 1) Bx 0 ~ e 0 dx = B • (41.19) Substituting this in (41.18), we have an expression for the rotational energy of a diatomic molecule or any linear molecule -0 RT_ N (41.20) We note that the concepts of “high” temperature for vibrations and rotations do not coincide in the least. With respect to the rotational .specific heat of oxygen, the temperature must be higher than 10° K to be regarded as high, while with respect to vibrational specific heat, it must be above 2,000° K. Therefore, in a very wide range of temper¬ atures, in particular at room temperature, the specific heats of diatom- ic gases are constant, and consist of a translational part B and ^ 5 a rotational part equal to B, so that the total specific heat is y B. It may be seen by numerical computation that the rotational specif¬ ic heat does not tend to a constant limit monotonically, but passes through a maximum at 6 = 0.81 B, equal to 1.1 B. The rotational energy for a polyatomic molecule will be calculated in Sec. 47. Exercise Find the rotational energy of para- and ortho-deuterium. Particles with integral spin have a symmetric wave fimction. Let us now consider a system of two particles with integral spin, for example, a deuterium molecule. For comparison we shall also take two particles with spin zero. The .spin function of the latter is identically equal to unity; therefore their orbital 456 STATISTICAL PHYSICS [Part IV wave function can bo only symmetrical. With respect to the rotational function, interchanRo of the nuclei is equivalent to a reflection at the coordinate origin. Hence, if the spin of a deutoron were zero then the spectrum of molecular deute¬ rium would show the linos, coiTcspoiiding to odd rotational quantum numbers, to bo absent. In actual fact they exist in the deuterium spectrum, and the weight of stat(!S with even K is twice as great as for those with odd K. This is seen from the relative intensity of spectral lines that correspond to transitions from the appropriate states. W(( shall show 1 ,hat for a deutoron spin of unity, the weight of the ortho¬ states turns out twice the weight of the para-states. A spin projection of unity takes on three values: I, 0 , - - 1 . We denote the spin wave functions (of both douterons) that correspond to these projections as ( 1 ), ( 0 ), (— 1 ) ’t'z (0), (—1). Let us form all the spin wave functions of deuterium that correspond to a total spin projection 0 ; wo shall only take symmetric and anti¬ symmetric combinations: Symmetric functions Antisymmetric functions ( 1 ) (-- 1 ) ■! (- 1 ) ( 1 ), ( 1 ) (- 1 ) - (- 1 ) ( 1 ) . 'I'l ( 0 )+ 2 ( 0 ), For the total spin projection 1, we obtain i., (1) (0) -I- (0) (1) , 4,1 (1) 4, (0) - (0) 4, (1), (~ 1 ) 42 (0) + 4i (0) 42 (- 1) , 4i (- 1) 42 (0) - 42 (- 1) 4i (0) • And for a total projection ± 2 wo have 4i (1)42(1). 4 i(-1)42(-1)- The symmetric state has a maximum spin projection of two. Hence, the state for which the spins are parallel is symmetric. But there are six symmetric spin wave-function projections in all, and spin 2 has 2-2-t-l = 5 projections. Hence, of the functions with zero resultant projection, we can construct one function corresponding to a zero projection of spin 2. The other function with zero residtant projection corresponds to a resultant spin 0. In all, deutorumi has six ortho-states with a symmetric spin wave fimction. A spin unity has states given by an antisymmetric spin function because the maximum spin projection in those states is equal to unity. Thus, there are three para-statos. An oven rotational function of a deutoriiun molecule corresponds to the ortho-states, and an odd rotational function corresponds to the para- states. Then the total function is symmetric, as the ca.so should be for integral particle spins. The weight (duo to spin) for the ortho-states is six and for the para-states it is throe. Therefore, the statistical sum of ortho-deuterium is B KCfC-t I) (iJJ {2K + l)e ® , A'-=0,2,4,... and for para-douterium it is equal to BKtK+ 1) K = 1,3,5,... Sec. 42] THE APPLICATION OP STATISTICS 457 Here the equilibrium state at absolute zero is the ortho-state. The energies of both states [see (41.16) and (41.17)] are _ 6B ^ortho ^ • ®para Si 7? (2 -f 28 e ® ) . Compared with hytlrogen, the ortho- and para-states are interchanged here. (7ose to absolute zero, the basic contribution to the specific heat is given only by the ortho-state. Two thirds of all the moleoulos in equilibrium deuterium occur in this state at room temperature. Therefore, the rotational specific heat of rapidly cooled deuterium is loss than that of equilibrium deuterium at the same temperature in the ratio 2/3. Thus, by measuritig this ratio we can show that the spin of a deuteron i.s equal to >mity and not zero. Sec. 42. Tbc Application of Statistics to the Electromagnetic Eield and to Crystalline Bodies The statistical equilibrium of matter and radiation. lu this section we shall first of all consider radiation in a state of statistical equilibrium with matter. The conditions for such equilibrium are achieved inside a closed cavity in an opaque body. The walls of the opaque cavity absorb radiation of all frequencies and hence they also radiate all frequencies: if a direct quantum transition is permissible, then the reverse transition is also permissible. Therefore, radiation arrives at a statistical equilibrium with matter, that is, in unit time there is an equal amount of absorbed and emitted energy of electromagnetic radiation per unit surface of the cavity for every direction, frequency, and polarization. An equilibrium density of radiation energy is thus set up in the cavity. It can be shown that in this case temperature of radiation is equal to the temperature of the walls. The necessity of this will be es])ecially clearly seen in the sections dealing with the fundamentals of thermodynamics (Sec. 46 and 46); for the time being we shall merely note that it is natural to regard the temperatures of systems in equilibrium as identical. The absolutely black body. Equilibrium radiation can be experi¬ mentally studied by making a small aperture in the wall of the cavity: if it is of sufficiently small dimensions the equilibrium state will not bo noticeably changed. Radiation incident on such an aperture from outside the cavity is absorbed in it and does not get outside. In this sense the aperture resembles a black body which does not reflect light rays. For this reason it is called an “abaolvte black body," and the equilibrium radiation coming from the aperture is called “black-body radiation." 458 STATlSTIOAIi PHYSICS [Part IV This term is somewhat paradoxical since it contradicts the obvious picture. Indeed, an absolutely black body in equilibrium radiates more than a nonblack body because it absorbs more, and in equilib¬ rium the radiation and absorption are equal. If a body having a cavity and aperture is brought to an incandescent state, the aperture will exhibit the brightest glow. The statistics of an oscillator field representation. Planck’s formula. In this section we shall consider the application of statistics to equilibrium radiation. For this it is necessary to quantize the radiation. Unlike the statistics of a gas, the statistics of radiation does not permit a limiting transition to equations, with the quantum of ac¬ tion being eliminated entirely. This will become clear a little later. In quantizing the field, a double approach is possible. Firstly, a field may be represented as a set of linear harmonic oscillators by characterizing each oscillator with a definite wave vector k and polarization a (a — 1, 2). It is obvious that all these oscillators are difierent (as to their k and a). The quantum properties of such os¬ cillators are not apparent in calculating the number of states of the field; their only manifestation is that the energy of each of them cannot be equated to an arbitrary number, but belongs to an oscil¬ lator-energy spectrum; i.e., equal toh and the number of oscil¬ lations with frequency o> is, according to (25.24), J , , Fto® dci / jn dg(0i)== . (42.2) Here, in contrast to formula (25.24), both possible polarizations of oscillation with a given frequency are taken into account, and K = to/c has been substituted. Hence the energy of an electromagnetic field in the frequency interval dco is d if (w) = Vha^ do) (42.3) The radiation spectrum of the sun is close to this frequency distri¬ bution. The statistics of light quanta. Let us now approach formula (42.1) from another direction. We have said that the electromagnetic field is viewed as an assemblage of elementary particles—flight quanta. Sec. 42] THE APEUOATION OV STATISTtOS 469 Quanta of the same frequency, direction, and polarization are in¬ distinguishable from one another. Therefore quantum statistics are applicable to them as to particles. At the same time quanta have integral angular momenta; this was mentioned in Sec. 34. Therefore they are not subject to Pauli exclusion, and possess a Bose and not Fermi distribution. But, as opposed to gas molecules, which are subject to a Bose distribution, the number of quanta is not a constant quantity, since quanta may be absorbed and radiated. This is why the supplementary condition (39.12) does not apply to quanta. It is easy to pass from the general Bose distribution to a special case, when condition (39.12) is not imposed; for this it is sufficient to put equal to zero the parameter (x, by which equation (39.12) is multiplied (fx was introduced to satisfy the' condition N =const). Then the Bose distribution is simplified: n= . (42.4) 6®-l Taking into account that for a quantum z = h(>i, we once again obtain (42.1). Thus, formula (42.1) denotes either the mean vibra¬ tional quantum number of an oscillator in an assembly subject to Boltzmann statistics, or the mean number of light quanta subject to Bose statistics. As we have already said, certain oscillators obey Boltzmann statistics; they are difierentiated by the numbers n^, rig, a (see Sec. 27), while the statistics of distinguishable particles is nonquantum. Let it be recalled that we differentiate between quantum and nonquantum statistics according as the particles are distinguishable or not. The impossibility of the limiting transition h -> 0 in the statistics ol the electromagnetic field. Let us now turn, for a time, to the oscil¬ lator picture. On classical theory, the mean energy of an oscillator is equal to 6 [see (41.11)-(41.13)]. If we multiply it by dg (w), the classical Rayleigh-Jeans formula for the energy of equilibrium radiation results. d^(w)cias. = ^J^0. (42.5) But this formula is obviously inadequate for large frequencies; upon integration with respect to to it gives an infinite total energy. It was precisely here, in statistics, that the classical representations first so obviously failed. Therefore, in 1900, Planck proposed for¬ mula (42.3); it was here that the quantum of action appeared for the first time in physics. Formula (42.6) is correct only for frequencies that satisfy the inequality 460 STATISTICAL PHYSICS [Part IV The total energy of equilibrium radiation. It is easy to find the total energy of equilibrium electromagnetic radiation from formula (42.3). Integrating with respect to w, wo obtain S-- oo Vh r J e«--l 0 h* r dx J • (42.6) 4 I’he integral in (42.6) is merely an abstract number, equal to^ (see Appendix, p. 586), so that the required energy is proportional to the fourth power of the absolute temperature (the Stefan-Boltz- mann law). •*—»»> Radiation from an absolutely black body. The result (42.6) can be verified from the emissivity of an “absolutely black body.” It is easy to rebate it to the energy S’. For this it is sufficient to calculate how many quanta fall from inside in unit time upon unit surface of a cavity, normal to the surface. We have indicated that if we take away a small section of the wall, radiation will pass tlirough the aperture with the same composition as that falling on the wall. The velocity of each quantum is c, so that its normal component is equal to c cos 0, where 6^ is the angle with the normal. In miit time these quanta will strike a square centimetre of the wall from the whole volume of a cylmder with base 1 cm* and height c cos 0. The energy included in the volume of this cylinder is equal to -pr • c cos h. Tlic fraction of quanta flying in unit solid angle is equal to , so that the total energy falling on a square centimetre of the wall in unit time is TC 27T T —~ J d 9 I sinO-dS- • 0 6 ccosh a- r f ~4~V 7r“ O'* -r'^ h* __ A- ■ 60 c=>/t» ■ (42.7) The constant in front of T* is equal to 5.67 x 10~® erg/cm* sec • deg*. Formula (42.7) cannot be directly applied to an incandescent solid body without ascertaining to what extent it may be regarded as black. Due to the fact that the sun’s luminous shell (chromosphere) is nearly opaque to radiation, the spectrum it emits is close to the equilibrium spectrum (42.3), even though it does not exactly coin¬ cide with it. The temperature of the chromosphere, as determined from (42.3), is approximately 5,700°. The pressure of equilibrium radiation. It is also easy to calculate the pressure of equilibrium radiation. It is convenient in doing so to apply the same reasoning that led to formula (42.7). Now, however. Soc. 42] THE APPLICATION OP STATISTICS 461 instead of calculating the number of quanta, it is necessary to cal¬ culate their normal component of momentum transmitted through a square centimetre of surface. This component is equal to the quan¬ tum energy Ato divided by c and multiplied by cos Therefore, unlike formula (42.7), we must integrate cos®^ instead of cos 11. In addition, for every incident quantum in the equilibrium state there is a similar quantum radiated in the reverse direction, so that the transferred momentum is doubled. Whence the pressure is 2 Tt Jt/2 0 0 i.c., one third of the energy deiLsity. The same woidd be obtained from the derivation of equation (40.22) if the momentum were put equal to e/c instead of m.v. We note that in Lebedev’s experiments, where the pressure of a directed beam was measured, and not of light arriving uniformly from all directions, p = ^/F; the pressure of the directed beam is equal to the energy density without the factor 1/3 (see Sec. 17). From (42.8) and (42.6), the pressure of electromagnetic radiation increases in proportion to the fourth power of the temperature while the gas iiressure is proportional to the first power. Therefore, radiation pressure will always predominate at a sufficiently high temperature. At high temperatures the pressure of a substance can always be calculated from the ideal-gas formula, because the interaction energy between particles becomes small compared with their kinetic energy. Hence, JVO p=—• By considering that atoms are dissociated into nuclei and electrons, it is easy to express the ratio NjV in terms of the mass density. Let us suppose that the substance consists of hydrogen. Then for every proton there is one electron. If the density of the substance is p, then the ratio NjV is 2plm., where m is the mass of a proton and the factor 2 takes into account the electron. This gives (42.9) From (42.8) and (42.6), the radiation pressure p, is From this we obtain the relationship between density and tem¬ perature when the radiation pressure becomes equal to the gas pres¬ sure: 462 STATISTICAL PHYSICS [Part IV For example, for a density p = 1 gm/cm®, both pressures become equal if the temperature is equal to 4 x 10’ deg. Radiation pressure is important in the interiors of certain classes of stars. The frequency corresponding to the maximum radiation-energy density in a spectral interval dw. The maximum energy in the distri¬ bution occurs at a frequency determined from the equation - d<3i \ Performing the differentiation, wo have (42.11) _h <.>o 1 - C ht,i„ 30 * This equation has a single solution with respect to ft (42.12) ftOig ~ir 2.822. (42.13) Thus, the frequency corresponding to maximum energy in the spec¬ trum of black-body radiation is directly proportional to the absolute temperature (Wien’s law): 2.8220 “0 — A (42,14) We notice that the numerical coefficient in the formula would have been different if we had considered the wavelength distribution instead of frequency distribution (see e.xer- cise 1). It is interesting to note that the corresponding Avavelength in tlie solar spectrum is very close to that for the maxi¬ mum sensitivity of the human eye. The curve of the distribution h 6> e ® —1 is shown in Fig. 48. Spontaneous and forced emis¬ sion of quanta. At the beginning Fig. 48 of this section we pointed out that thermal equilibrium between atoms and radiation is attained in a closed cavity. The presence of atoms capable of radiation and absorption is necessary m general in order that the radiation may arrive at equilibrium; this is because Sec. 42] THE APPUtOATION OP STATISTICS 463 separate oscillators, corresponding to normal oscillations of the electromagnetic field, are completely independent of one another, and any initial nonequihbrium distribution is maintained until there is an exchange of quanta via absorbing atoms. In Sec. 34 we derived an expression for the probabfiity of fight emission by an atom. According to (34.46), the radiation probability in unit time is (42.15) We shall now consider atoms which are in thermal equilibrium with matter. Let the frequency Wi, satisfy the relationship AtOio=ej—E q, where and Eq are the energies of two atomic states. In equilibrium, atoms with energy e^ radiate as many quanta with frequency Wm as are absorbed by atoms with energy Eq. In accordance with the principle of detailed balance, the probabili¬ ties for direct and reverse transitions are connected by the following relation: !7i^io “ ^0^01 • (42.16) Indeed, the first-approximation formula of perturbation theory (34.29) is applicable to radiation and absorjition processes, since the interaction of matter with radiation may be regarded as weak. From this formula, the probabilitie.s for the transitions 1 >0 and 0 ->1 are, respectively, W^oi = 4^l^oil=‘?,- (42.17) But according to the Hermitian condition (34.15), the squares of the moduli of the matrix elements | M’qi and | |* are the same so that if we multiply expressions (42.17) by the weights of the initial states, the result will be equation (42.16). The formula for the probability of absorption related to the case when a single quantum of frequency toi^ existed in the field before absorption. If there were n (wm) such quanta before absorption, then it is natural to assume that the probability of absorbing one of them in unit time is n (wjo) times greater. This assumption is justified in electromagnetic-field quantum theory. We shall therefore assume the probability of absorbing in unit time one of the n (ojo) identical quanta in the field to be equal to n (wm) ^olFoi. In accordance with the principle of detailed balance we must have the same probability for the reverse transition, i. e., the emission of a quantum by an atom occurring in state 1 when there are n ( to^o)—■ 1 such quanta in the field; this is because the transition is reversed with respect to the one just considered. We represent both transitions thus: 464 STATISTICAI. PHYSICS [Part IV quantum absorption quantum radiation Thus, in accordance with the principle of detailed balanee, the prob¬ ability of emission of a quantum must likewise be which can also be represented as [(n — 1) + 1] !7i IFm- Because of equation (42.16) the probabilities for both direct and reverse transitions will be equal. Hence, if n --1 quanta exist in the field, then the probability of emis¬ sion is proiJortional to n, i.e., to the number of quanta increased by unity. If, for exanijile, there were no quanta in the field before emission, this factor of jirojiortionality is equal to unity. In this case the emis¬ sion is termed spontaneous. But when there are quanta in the field, they stimulate, as it were, further emission of quanta with the same frequency, direction of jiropagation, and polarization. The emission produced hy them is called forced. The existence of forced emission can also be proved by means of quantum field theory, just as the pro¬ portionality factor n in the absorjition jjrobability. The idea of forced emission was introduced by Einstein. The derivation of Planck’s formula from the relationship between the quantum emission and absorption probabilities. Let us now consider atoms in thermal equilibrium with an electromagnetic held. Let the quantity n (wio) denote the equilibrium number of quanta. The condi¬ tion of statistical eipiilibrium is that atoms occurring in state 0 absorb as many quanta with frequency in unit time as are emitted by atoms in state 1. Then the number n (wjo) docs not change with time, i.e., equilibrium is attained. The number of acts of absorption by all the atoms in unit time (from state 0 in which there are atoms) is equal to ■^0 ^^01 ^ (^lo) • (42.18a) The number of acts of emission by all atoms in state 1 in unit time is JViIPiof«(inax — (42.32) Condition (42.31) is selected so that at high temperatures the correct law ^ = 3 NQ is automatically obtained. At medium temperatures 6 k= -^^?*-i8 substituted as the upper limit in the integral (42.29) in place of oo, so that the energy expression has the form ^max (42.33) Changing to the integration variable x — and denoting Awjnax = =0D, we can rewrite the lattice energy thus: Sep. 42] THE APPLICATION OF STATISTICS 471 271* \«; u) I /t» J € ' a:* dx T (42.34) At low temperature 0 d > 6, so that the upper limit in the integral is replaced by infinity. Then the integral is equal to , and for the energy we have The exact formida (42.30) assumes the same form if wo replace in it by Vt and ui, which are independent of direction. We shall now show how to determine 0 d from experimental data on specific heat and, independently, from elastic constants. The following values of specific heat G are known for tungsten (from the data of F.F. Lange): T —26.2°K, C —0.21 cal/mol • deg.; T = 3S.0°K, C = 0.75 cal/mol • dog. The cube of the temperature ratio is equal to 3.37, and the ratio of specific heats is 3.68. We may assume that in the given temperature range the T® law for specific heat holds. Substi¬ tuting Or) = Acoma.'c in formula (42.35), we determine Wmux with the aid of (42.32). This gives 12 Converting this to heat units, we write Here R = 1.96 cal/mol • deg. Substituting the specific heat at the lowest temperature, we find I’d = 340°. We now determine To by proceeding from the elastic constants for tungsten. We have to give, without derivation, the formulae which connect u, and ui with the shear modulus and the bulk modulus for tungsten (see L. D. Landau and E. M. Lifshits, The Mechanics of Continuous Media, Gostekhizdat, 1953, p. 744 or A. Love. A Treatise on the Mathematical Theory of Elasticity, Ch. XIII, Cambridge, 1927). Here, K is the bulk modulus, which, for tungsten, is about 3.14 x 10^® dyne/cm® at low temperature. 0 is the shear modulus equal to 1.35x10®® dyne/cm®. The density of tungsten is p = 19.3 gm/cm®. Hence,«; = 6 x 10® cm/sec, = 2.64 x 10® cm/sec. For tungsten the ratio NjV is equal to 0.635 x 10®®. Whence, if we calculate it from (42.32), 14. 4 Cl V, lAi^ 1 Am 4.61 X 10'“ X 1.05 X 10-” oeoo Wmax IS equal to 4.61 X 10®®sec-®andTD=- 1 38 x 10 ^ ’*-= 352 . 472 STATISTICAL PHYSICS [Part IV The agreement with what waa obtained from specific heat turns out to be even better than could have been expected, because the elastic constants do not strictly refer to the temperature at which the specific heat was determined, and also because tungsten is a crystalline substance and its elastic properties are characterized by three moduli of elasticity instead of two (see Landau and Lifshits, loc. cit., p. 675). For a number of substances we have the following values of Debye temperature To- Pb — 88°, Na — 172°, Cu — 315°, Fe — 453°, Be— 1,000°, diamond—• 1,860° (all from absolute zero). At high tomi)crature, 0 > Od, we must put — 1 x, so that K_/ 2 1 \ 0-Oj^ StcS I u] 0 = 3iV6, (42.36) which is wliat we demanded. For 0 6 d, formula (42.34) agrees with experiment qualitatively. Wo note that we must not expect complete agreement, because the initial assumptions made in deriving this formula are not quantitative in character. It is not worth the attempt to make formula (42.34) more accurate, without taking into account the exact form of the dependence of w upon k. The attempts at correcting this formula, which are sometimes made, are simply in the nature of adjustments. Exercises 1) Write down the fornnila for the wavelength distribution of black-body 27rc radiation energy. Proceeding from the fact that o (X) - — -V X^e -l) The nnvximum is debned by the C(juation 2 7t:7ic X---== 4.90.5. , wo have 2) Show that if lio.se particles interact with a Boltzmann gas the probability of a pai’ticlo appearing in a certain state is proportional to n -t-1, where is the number of particles already in that state, and the probability of a particle disappearing is n. Lot the energy of a Boltzmann particle be e and that of a Bose particle, v). Lot ns consider the process in which there occurs the transition e Tf) -> e' -t- T)' i.e., the interaction of these particles changes their initial state with energies e. I) to a state witli energies z' and rj'. In statistietd equilibrium we must observe the balance IPee' Ac Ut) (1 -r n,)') “ TV z'z V e' (1 W,) where TVee' is the probability of direct transition and TFe'e is the reverse- transition probability. Putting Sec. 42] THK APPLICATION OF STATISTICS 473 n-g g Nz-=ge6 ® . Ns’==gt'e ® , / 11-1^1 \-i / -n' - ii, y -1 «„=\6 ® - 1/ , >iV=\e ® — 1/ , we see that the balance equation is satisfied if We z' — Wz' z. For simplicity wo have put gz — gz’- The presence of spontaneous emission is duo to the Bose distribution. 3) Find the total number of quanta in black-body radiation at a given temperature N ■■ V Further (see Appendix), I ha Tt^/FC® I J e 1 J 0 0 oo (lx >—r Hence, ri-10 e^ — ] ti - I 3 OO CO OO e-"^ x= (lx = 2]’ J c ” !J^ H = ' H = 1 0 »-l The sum is approximately 1.2, so that JV- 2.4 2 TC“ V(P 'h^ (y< ■ 4) The atoms are situated in the form of a linear chain. Wo shall denote the disjfiacoment of the nth atom by a„. The force acting between the nth and (n.-f- l)th atoms is equal to a (on-t-l— On). Find the equations for the vibra¬ tions of the chain. Ignore the interaction between the more distant neighbours. The vibration equation for the nth atom is ma„ a {a„ + i -f a„_i — 2a„). We look for a„ in the form (In ~ h (t) e'f". Substituting this in the initial equation, we find, after cancelling eF" tni) (t) = a}) (t) (e'f -b e“F — 2) == 2 a}) {t) (cos / — 1) ^ — 4 a sin* ■ b (t) > so that the oscillation frequency for a given value of / is If the distance between the atoms is d then n = -^, where x is the equilib¬ rium position of the nth atom. Putting = fc, we have e‘^" = e’*'*, so that / can be called the wave vector, considering that the length is measured in imits of d. For small /, as was asserted, the frequency is proportional to f ; 474 STATISTICAI, PHYSICS [Part IV Sec. 43. Bose Distribution The choice of sign of |x. The Bose distribution has very peculiar properties at low temperatures. We shall suppose that the atoms do not have spin; such, for example, are helium atoms with atomic weight 4. Both the electrons in the cloud of the helium atom and the protons and neutrons in the helium nucleus are in the l«-state. They all go in pairs and by the Pauli principle the spins are antiparaUel. Therefore, the resultant spin is zero. From (39.30), the weight of the state of a spinless particle is d(j (e) -= I’«i% s/tdz (43.1) The normalization condition (39.23) looks like (43.2) This condition can be satisfied only for negative (x. Indeed, if we suppose that jx is greater than zero, then the denominator of the g - t* integrand will be negative for e < |x because then e * < 1. But this is impossible because the distribution function is, by its very meaning, a positive quantity. Hence, ;x<0. At high temperatures the Bose distribution ])asses into the Boltzmann distribution in accord with (40.6). The sign of . As the temperature diminishes, (x decreases in absolute value. This can be shown generally with the aid of (43.2). Differentiating this equation as an implicit function wo have The integrands in (43.3) are essentially positive quantities [(e — (x) >0, because tx<0], and therefore <0. Hence, as 0 decreases, the absolute value |(xl diminishes monotonically since [x must increase. Sec. 43] BOSE DISTRIBUTION 475 We shall now show that [x becomes zero at a temperature other than zero. To do this we put (i, = 0 in (43.2) and find the corresponding value 0 = 0o: CO VmVt i \tdz J 0 V m’lt 0„*/» / V-'’ ^ Ar "2V27rVi»“' I e'“ 1 ~ • 0 (43.4) The integral simply represents an abstract quantity: it is equal to 2.31 (see Appendix). Therefore equation (43.4) is satisfied by a value of 0^ that is different from zero. Bose condensation. What will happen when the temperature is reduced further ? p cannot go from negative to positive values since, as wo have shown at the beginning of the section, this would lead to negative probability values, p cannot become negative once again, because is always less than zero so that p varies only monotonically, if it is at all capable of varying. Therefore, the only possibility is for p to remain equal to zero after it has once attained its zero value. But then equation (43.2) is no longer satisfied if the temperature is less than Oq, and N does not change. On the contrary, it can be seen from (43.4) that if we define the number of particles as CO 2.31 Fm7,(rt, 2'UTz‘‘h^ (43.6) for 0<0o, it decreases with the temperature in proportion to fi’/s. What happens to the remaining particles which number N — A' ? As opposed to light quanta these particles cannot be absorbed. Therefore, they will pass into a state which is not taken into account in the normalizing integral (43.2). The only state of this kind possesses an energy equal to zero: due to factor Ve it does not contribute anything to the integral (43.4). In normalization we can isolate the particles occurring in the zero state in a separate term. If a finite number of particles go to the zero-energy state, they will naturally fall out of the integral. N' particles remain continuously distributed, but with the value p = 0. Thus, at a temperature 0<0o, the whole distribution consists of an infinitely narrow “peak” at e = 0 and of particles distributed according a (e ® — l) law. At absolute zero all the particles are in a zero state: this state of a Bose gas is obviously defined uniquely. It will be noted that a Boltzmann gas would behave in an entirely dififerent way when the temperature tended to zero. 476 STATISTICAL PHYSICS [Part IV Liquid helium. Helium with atomic weight 4 obeys Bose statistics since the spin of its nuclei and of tlie electronic shells is equal to zero. It is therefore interesting to see whether anything like this “Bose condensation” is observed in helium. It is difficult to give a unique answer because at low temperature helium is a liquid, and the Bose distribution, which relates to an ideal gas, does not apply. Nevertheless the qualitative aspect of the result obtained for a gas may still hold. Namely, it may be supposed that at a certain temperature part of the gas will pass into a zero energy state and, accordingly, will not contribute to the specific heat. Liquid helium does, in fact, experience a peculiar change of state at a temperature of 2.19° K (at atmospheric pressure). S]ieaking of a monatomic liquid, which is what liquid helium is, it is difficult to imagine any change of state related to a rearrangement of the atoms in space. Therefore, it is interesting to compare the actual temperature of transition in liquid helium with the tem])orature at which Bose condensation would occur in gaseous helium of the same density. The density of liquid helium is equal to 0 . 12 gm/cm*. Whence thcratio ^ — ^ 4 ^ x 6 x 102® = 0.18x 10^®. Consequently, according to (43.4) the temperature Oq is 0 „ - 0.18 • .0.86 • l. tl . 1.18 . 10 2.:U • 17.1 • 10 - 3 « : 5.86 . 10 " i.:i8”io i« 7' - ' 0 r- 2 . 8 °, 3.86- 1()-1«; which is close to the transition temperature. At the transition, the specific heat of helium experiences a discontinuity. In the case of a Bose gas, only the derivative of the specific heat with respect to temperature has a discontinuity. Superfluidity. P. L. Kapitsa discovei’cd that below the temperature of phase transition, liquid helium possesses a most remarkable prop¬ erty : it is capable of passing through the finest slit without exhibiting any signs of viscosity. This property was called superfluidity. L. D. Landau developed a theory of superfluidity proceeding from the supposed quantum-level spectrum for a liquid. On the basis of this theory he built the hydrodynamics of a superfluid, which differs from conventional hydrodynamics in that each point possesses two velocities instead of one: a normal and a superfluid component. The occurrence of two velocities means that in a superfluid two types of sound vibrations may be propagated: ordinary sound, in which pressure and density oscillate, and “second sound,” which is connected with the relative motion of the normal and superfluid components. The second sound was demonstrated in an experiment carried out by V. P. Peshkov using a method proposed by E. M. Lifshits. The Soc. 44] FERMI DISTRIBUTION 477 experimentally found velocity of second sound (which is small com¬ pared with the velocity of conventional sound) is in excellent agree¬ ment with Landau’s theory. The question of thq relationship between superfluidity and Bose condensation cannot be considered fully resolved. It may be suggested that the superfluid component corresponds to that part of the helium which has passed to the zero state. This hypothesis is strongly sup¬ ported by the fact that the liquid isotope of helium with atomic weight 3 is not superfluid: the nuclear spin of helium 3 is equal to -i-, so that its atoms are subject to the statistics of Fermi and not Bose. Accordingly, they cannot aU pass into the zero state together: the Pauli principle does not permit this. N. N. Bogolyubov showed that a gas which is close to an ideal gas and consists of Bose particles possesses an energy spectrum which, according to Landau’s theory, a superfluid liquid should have. How¬ ever, no one has so far succeeded in proving theoretically that it is precisely liquid helium below the transition point that should possess such a spectrum. Exercise Calculate the onorgy and pressure of a Boso gas below tho transition point. Kor tho energy wo have CO 0 (see Appendix). The pressure is determined from the general relationship (tO.22): _ 2 <# _ l-lS/a’/jO'/s Thus, tho pressure of a Boso gas below the transition point is independent of volume ami depends only upon the temperature. If wo compress such a Boso gas its particles will go to tho zero-energy state. Conversely, upon expansion tho particles will come out of the zero-energy state until there are none left. If expan¬ sion continues tho pressiu-e will begin to decrease. Sec. 44. Fermi Distribution The form of the Fermi-distribution curve and its interpretation. The criterion for tho transition from quantum statistics to classical statistics is that [see (40.7)] If the inequality is reversed, then essentially quantum properties of the statistical distribution appear. In this section we shall consider 478 STATISTICAL PHYSICS [Part IV the properties of the Fermi distribution when the inverse inequality N ^ 3(0) I in 0 T ^ I* or an equivalent inequality (44.1) (44.2) is satisfied. From (30.26) and (39.30), the Fermi-distribution curve is of the following form: dn{e) - K (2m®)'/ieVt(le (44.3) Here, a weight factor 2 is introduced, since we have put j — y ' first factor in (44.3) represents the total number of states between e and e -( de, while the second factor represents the probability that these states are occupied. We can interpret the function /(e) = ! 1 (44.4) as a probability and as the mean number of particles, because / (e) is contained between zero and unity. A similar function in the Bose distribution could only denote the mean number of particles in one of the quantum states with a given energy, because the Bose-distri- / V 1 bution function \e ® — 1 / is sometimes even greater than unity and must not be interpreted as a probability. Let us see how the curve / (e) behaves when > 1. When s = 0 we obtain /(«)=-—,— - = 1. e ® + 1 JiL e- ti because e ® is a small number. The quantity e ® is also a small number as long as e remains smaller than [x, while / (e) is close to e- |i unity, like / (0). Only when e — [x is comparable with 6, is e ® of the order of unity, so that / (s) begins to decrease noticeably with further increase of e. For e = ix, / (fx) decreases tOy: = -go^ri = 2 • Sec. 44] FBRMI DISTMBimOJSr 479 For still greater values of e, / (e) decreases exponentially because unity can then be neglected in the denominator, and, for e > (i, / (e) becomes the Boltzmann distribution \JIS1 /(e) ~e ® . The Bose distribution also has the same limiting form. The curve / (e) is roughly shown in Fig. 50. The region s, where / (e) changes from unity to zero, has a width of the order 0, since ^ T~ is comparable with unity only if e - for smaller s the exj)onential is con¬ siderably smaller than unity, while for larger e the exponential is considerably greater than unity. Fermi distribution at absolute zero. We shall call the region of transition of / from unity to zero the spread region of Fermi distri¬ bution. As the temperature decreases the spread region narrows and, at absolute zero, becomes a shar]) discontinuity /, so that the distribution function takes the form of a right angle. Fig. 60 shows this step by a broken lino. The value of (ji at absolute zero is called (Xq. Hence, at 6 = 0, all states with energy less ■ than po are occupied with unity prob¬ ability (i.e., with certainty), while those with energy greater than are empty, also with certainty. This result can likewise be obtained directly from Pauli’s principle without resorting to statistics. From (39.32), a definite interval of momentum-component values £\px, Apy, Apz corresponds to one state of particle motion. If the particle is contained in a box with sides %, Uj, aj, then it follows from the uncertainty relation (23.4) that hioiy Fig. 50 Ap* 271*. Apy 2 Tzh I^Pz 2 Tzh since these quantities show by how much the momentum components of two particles must differ in order that the particles may be regarded as occurring in different states of motion. This follows not only from the uncertainty relation, but can also be seen strictly when computing the states leading to formulae (26.23) and (39.32). Here, each state must be identified not with the volume of the parallelepiped, but with one of its vertices whose coordinates are given by the three integers Ui, n^, n^. The coefficient 2n in the uncertainty relations is taken so that both definitions for the number of states agree. If we plot p*, py, pz on coordinate axes, then to each state of spatial motion of the electron there correspond three quantum numbers 480 STATISTICAL PHYSICS [Part IV Wi, Wj, n^. These quantum numbers specify the number of the parallele¬ piped with sides Ap*, Apy, Ap^. It is shown in Fig. 51. AU the space in which the axes p*, py, pz are drawn can be A ~jA filled with such boxes. Smee three quantum f l_ numbers correspond to a single box and, in '' 7ap addition, the state is also given b^y the spin, there may be two particles with .spin having momen¬ tum projections in the same interval Ap*, Apy, / l\pz. The spins of these two particles are anti- /p^ ^ parallel. ji'jg gj Thus, the space p*, fy, Pz may be divided into boxes or cells with dimensions ApxApyApz = (_27t/t)3 «3 Clr.h)^ V ’ (44.5) where there are no more than two jiarticles in each cell. The closer the cell to the coordinate origin, the less the energy it possesses, because the energy is equal to e = (pj -f pj -1- pi) . In other words, it is proportional to the square of the distance of the cell from the origin. Let us now consider the state of a gas at the absolute zero of temper¬ ature. If the gas consisted of only two particles, then at absolute zero the states of both particles woidd fill the cell closest to the origin. In accordance with the Pauli principle, the next two particles cannot enter the same cell: they are forced to take up positions further from the origin. As the number of particles increases, cells are filled which are situated further and further from the origin; but each time two particles are added they fall into a free cell closest to the origin, because, by definition, absolute zero corresjionds to the least possible energy of the gas as a whole. If there are very many particles, their cells will densely fill a sphere whose centre is the coordinate origin. All states inside the sphere are filled with unity probability, while those outside the sphere are free—also with certainty. The limiting energy of Fermi distribution. If we denote the energy corresponding to the boundary of the sphere by e^, then it can be seen from Fig. 50 that Sq —(Xq. [Xq is the limiting energy of a particle at absolute zero. It is very easy to calculate Sq or (Xq. Since at absolute zero the function / (e) is equal to unity for all e < [Xq, the total number of particles N is, from (44.3), N r(2TO^)Vi j\/ -j V{2mfItem’ll - J ^ 3 > 0 (44.6) Sec. 44] FEBMI DISTMBUTION 481 whence So = 3’/.7t‘/, 2m \v) • (44.7) The same can be seen without the aid of / (s). Indeed, the radius of the sphere of greatest energy is Po = V2mso . Its volume is But this same quantity is equal to the number of filled elementary cells (with two particles per cell) multiplied by the volume of a single cell — - . Consequently, -|-7c(2mEo)’/» = N (2izhf 3 V ’ (44.8) whence equation (44.7) is again obtained. At absolute zero the state of a Fermi gas as a whole is defined uniquely: in quantum statistics it is necessary to indicate which states are occupied by separate particles, but it is impossible to deter¬ mine by which 'particles they are filled. In the given case all the states inside the sphere with limiting energy Sq are filled by particles. The criterion for the closeness of the Fermi distribution to the distribution at absolute zero (based on the form of the distribution). At a temperature close to absolute zero thermal excitation can be imparted only to those particles whose energy is close to eo = [i.o. Indeed, as long as 0'^So> ^ thermal excitation of the order 0 cannot be imparted to a particle whoso cell lies deep beneath the surface e = eQ, because the states between the surface and the given cell are occupied, and the energy 0 is insufficient to remove the particle beyond the limits of the surface boundary. Therefore, only those particles whose energy differs from by an amount of the order of 0 can take up free places. Deeper states will remam densely filled as before. Thus, the filling probability wiU be almost equal to unity for all energies e (44.11) Whence (TluF _ 3 dp r>2) s'/a m = 0.273 X 102" bai-i . (44.12) Ya. I. Frenkel noted that the compressibility of alkali metals is close to the compressibility of an electron gas. Sec. 44] FBBMI DISTRIBUTION 483 Indeed, expressing NjV in terms of atomic weight and density, we obtain the following table: Li Na K Rb Cs - X 10“ from equation (44.12) \ tip 4.7 13 ! 37 i 62 79 -Jr X 10“ from experimental data h cp 8 15 32 40 61 In a crystal lattice there are, of course, not only forces of repulsion between particles, but also cohesive forces. The equilibrium of these forces with the forces of repulsion determines the characteristic vol¬ ume which every condensed body, solid, or liquid has in the absence of external pressure. Ordinary atmospheric pressure gives a force which is negligibly small compared with these tremendous forces that keep bodies in their volumes. In order to change the volume of a body by only one per cent, pressures are required in the order of tens of thou¬ sands of atmos])hcrcs. The coincidence of theoretical and ex|)erimental data indicates that when alkali metals are compressed the cohesive forces change insignif¬ icantly comi)ared with the forces of repulsion. It is even conceivable that the state of the valence electrons in alkali metals is perturbed to a comparatively small degree by the atomic residues, and, to some ex¬ tent, is close to an electron gas. Compression affects but little the electronic shells of the atomic residues, and therefore the compressi¬ bility of alkali metals is close to the compressibility of an ideal Fermi gas. That this should bo so is, of course, not at all obvious beforehand. Paramagnetism of alkali metals. According to Pauli, the paramagne¬ tism of alkali metals can also be cxiilained on the basis of the concept of a free electron gas. If we place a Fermi gas (consisting of electrons) in a magnetic field, the energy of the electrons, whoso spins are parallel to the field, will be equal to ^while the energy of electrons with opposite direc¬ tion of spin will be equal to Therefore, if those electrons whose spin is antiparallel to the field reverse their spin directions, then the energy of the gas must decrease. But all the places inside the limiting- energy sphere are occupied; so for an electron to change its spin direc¬ tion it must come out of the sphere into a free cell. But this increases its kinetic energy. Equilibrium is established between electrons with spins parallel and antiparallel to the field when their total energies become equal. Indeed, if there occurred a further transition of elec¬ trons into a state with spin parallel to the field, the increase in their * Here the Bohr magneton is denoted by 3 instead of g, so as to avoid confusion with the distribution parameter g. 31* 484 STATISTICAL PHYSICS [Part IV kinetic energy could not be compensated by a reduction in magnetic energy. Let there bo n electrons which have changed their spin directions. N Then there remain -- n electrons with spin antiparallel to the field, N ^ while have spins parallel to the field. The limiting energies are determined from formula (44.8), where we must put-^ ± n instead N ^ of -g-. Whence we obtain the following expression for the limiting kinetic energy of both types of electrons: (27tA)2 2m ’ (44.13) and the equation for the total limiting energies is N Since the binomials can be expanded in a series as follows: IN , \’/. IN\'h IN\% ‘ (2^1 (2) ^(2) S 4?> w Substituting this in (44.14), we find the number of electrons which change their spin directions in the magnetic field: n = N^H 3'/. 2 It* 1 3 m (44.15) Each of these electrons contributes a term 2 p to the total magnetic moment of the whole gas, because its moment projection on the magnet¬ ic field has changed from — p to p. The magnetic polarization (that is, the magnetic moment of unit volume) turns out equal to Jlf = 2p-^ _ 3V3 mp® IN\'l 3 jj (44.16) while the magnetic polarizability a, defined as the coefficient of H on the right-hand side of this formula, depends only upon the density of the electron gas and not its temperature: Ttv. ip; • (44.17) Indeed, alkali metals have a paramagnetism independent of tempe¬ rature. Let it be recalled that in accordance with the results of Sec. 40 [see (40.53)] atomic paramagnetism gives a magnetic polarizability which is inversely proportional to the temperature. Formula (44.17) agrees satisfactorily with experiment. Sec. 44] FEBMI DISTBIBUTION 486 Diamagnetism of electrons. L. D. Landau has shown that the quan¬ tized motion of electrons in a magnetic field—this motion is similar to their classical motion in a spiral—leads, in a weak field, to the appear¬ ance of a magnetic moment equal to 1/3 of expression (44.16), and of opposite sign. The nature of this eftect is purely quantum; if we regard the motion of electrons as classical then the additional magnetic moment becomes identically zero (see Sec. 46, exercise 13). If is of the order of 0, then the polarizability does not depend monotonically upon the field and exhibits much oscillation as the field increases. The oscillatory valuation of magnetic properties is, in fact, observed in very many metals. The potential distribution in an atom. We shall now show how to find the general form for the electron-density distribution in atoms via the notion of a Fermi gas. To a certain approximation, the electrons in heavy atoms resemble a Fermi gas. However, it must be noted that each electron occurs in the inhomogeneous electric field formed by the nucleus and the entire eonfigurfition of the remaining electrons. Let us first of all consider a Fermi gas at absolute zero in a potential field of the form showui in Fig. 52. U = 0 for 0b. Then the limiting energy of the electrons must be the same for 0 because in the case of spherical symmetric¬ al charge distribution the action of all the electrons of a neutral atom balances the action of the nucleus. Accordingly, the potential would also be zero when r = ro, because potential is the field integral: r i. e., the integral with integrand equal to zero at r'^r^. CO Thus, the follov'^ing three conditions would bo satisfied at the point r — T^\ n(ro) = 0 .- 9 (^o)~ 0 , | =o|w—^.i.e., the electron den¬ sity). The density is proportional to the 3/2 7 )ower of the kinetic energy [see (44.8)], • Here, ffo==eo---e = 47 ( 1 . 125 ZV 5 . r) . (44.25) If the distance from the nucleus is expressed in terms of x, the elec¬ tron density distribution is the same for all atoms to which the statis¬ tical method is applicable, i.e., for all elements of large and medium atomic weight. But the same x denotes a geometrical distance inversely proportional to , as can be seen from (44.23). Therefore, in heavy atoms, the main part of the electrons is concentrated closer to the nu¬ cleus than in the lighter atoms. The accuracy of the Thomas-Fermi equation (44.24) is determined by the quantity as can be shown from a strictly quantum- mechanical derivation by using a quasi-classical approximation. There¬ fore, equation (44.24) cannot, of course, be applied to the very lightest atoms that contain few electrons. 488 STATISTICAL PHYSICS [Part IV Substantiation of the boundary conditions for equation (44.24). The integral curves of equation (44.24) begin at the point 4' = 1 for * = 0, and fall with increasing x, accounting thereby for the screening effect, i.e., weakening of the nuclear field by the atomic electrons. The dimin¬ ishing function may either pass through a minimum, without attain¬ ing (p = 0, and then begin to increase, or it may intersect the a;-axis at a certain point x—x^, or it may tend to this axis asymptotically. The first possibility must be rejected at once, because it results oo in an infinite total number of electrons proportional to J x'l* dx, 0 see (44.18) and (44.21) (if we take (oo)>0). It is impossible to cut off the integration at some Xq when > 0. since this would correspond to a limiting total energy not equal to zero. If we take the second possibility, then the total number of electrons *■0 has a finite value and will be proportional to J x'i‘ dx. The electron 0 density and, hence, the electric field of a neutral atom also, must, by definition, become zero at the point x = Xf„ since the nuclear charge in it is completely screened by electrons. In accordance with (44.21) the electric field will be determined by the expression ^ _ d‘ = -^(x-Xo) , **0 whence it follows that k— — 6, in spite of the assumption that i;>0. Hence, tangency of the integral curve with the a:-axis is impos¬ sible at a finite distance from the origin, and asymptotic tangency must be assumed. And the condition = 0 is automatically satisfied at infinity. The charge distribution in positive ions. In positive ions, the charge of all the electrons does not completely screen the nuclear charge. Sec. 44] FERra DISTRIBUTION 489 because, at the point where = the condition ^-^==0 should not be satisfied. The electron density distribution in an ion is given by the integral curves intersecting the a;-axis. The point of inter¬ section determines the radius of the ion Zg. The order for flllihg the electron shells. From the potential distri¬ bution in an atom, we can determine the values of Z for which d- and /-electrons first appear in the atom. We first of all note that the electron density distribution in an atom must be associated with the angular-momentum distribution of the electrons. As we have already indicated, the limiting momentum of eleetrons is proportional to the ^/g power of the electron density. Therefore, close to the nucleus, where the electron density is great, the limiting momentum is also great, while at large distances from the nucleus, the limiting momentum is small. But the angular mo¬ mentum of an electron is determined by the product of the momentum by the distance to the nucleus, and close to the nucleus it is small despite the large limiting momentum. At large distances from the nucleus the angular momentum becomes small—this time as a result of the smallness of the limiting momentum. Hence, somewhere at medium distances, the angular momentum attains a maximum which is larger, the greater the electron density. Therefore, in heavy atoms with a large electron density, we find larger values of angular momentum. In order to find the greatest values of angular momentum that arc possible for a given Z, we shall proceed from the classical expression for energy in a central field [see (5.7)] Pr‘ , 2 m ' 2mr^ r (44.26) We must put <^ = 0 for the boundary energy, in accordance with the basic assumption (44.18). Then, for the radial component of momentum we obtain the expression p,= 2mZe^p (44.27) We can substitute in place of M^. But since formula (44.26) is written to a quasi-classical approximation, a better result is obtained if we also take the quasi-classical approximation for M*. It can be calculated using the same methods as those for determining the energy eigenvalues from formula (29.18). To this approximation = A* |i -f yj **. We notice that + y) * differs from I (1 -[-1) only by a quarter. We write (44.27) in the following form: 490 STATISTTCAIi PHYSICS [Part IV («.28) Let us now express the factor r in the radicand in terms of the nondimensional quantity x according to formula (44.23). Then pr will be p, A |/l.778^=/.x^- {l 1)1 (44.29) For pt to be a real quantity, the radicand must remain positive in a certain interval of values x. But since x<\i = 0 when a: = 0 and x~oo this interval is finite and contains the maximum point of the function x tj;. The maximum is equal to 0.488. Thus, the whole inter¬ val in which pt is a real quantity is contracted into a point for the value of Zi at which 1.778 • 0.488 • .|. -1)^, (44.30) the curve y =1.778 7'^^ a;t]; being tangential to the constant straight line y- 1/1 A)^ It follows that a given value of I in an atom may occur when Z satisfies the condition Z--0.155 (2/ -I 1)3. (44.31) According to this equation, electrons having 1 = 2 will occur for Z==19, while /-electrons (/ —3) will occur when Z —53. There will be better agreement if we take the coefficient 0.17 instead of 0.165. Using the numerical form of the function il' can be shown that the d- and /-shells are formed mainly deep inside the atom, as was shown in Sec. 33. The approximate integral formula for the Fermi distribution. In conclusion, let us consider a Fermi gas not at absolute zero, but at a temperature other than zero yet satisfying the inequality (44.9). It is convenient first to derive a general formula for the integral of the Fermi distribution that holds for 0 6^. Let us take the integral oo (44.32) where y (e) is some power function, for example V e, etc. We integrate (44.32) by parts: Sec. 44] FBBMI DISTB.IBOTION 491 CO oo Y(0) li (44.33) Let us write the second factor in the integrand thus: V • (44.34) e 0 : l/ie “ I ll The denominator of tliis e.xprcssion is largo both for s :;a and for 11-6 6 - - |1 £ > (X. The exponential e ® is lai ge in the first case, while e ® is large in the second case. Therefore, the whole expression differs Jioticeably from zero only in a narrow range of values s, different from p. by an amount of the order of 0. Let ns expand the function y(s) within this range and lot us terminate the expansion Avith the second term. y(E) = y(p) + (e- p)y'(p) + y" ([J-) ■ (44.35) We substitute this expansion in (44.33). Taking into account that the second factor in the integrand is very small for £ = 0, wo can perform the integration to s =^= — oo without making any perceivable n error. In addition, we shall neglect the quantity e * in the hitegrated term of (44.33). From this we obtain CO I ^ _ ,^(0) _ y (p) / d£ J—A ./ U " + 1 , + + y' 0 CO — oo 20 (44.36) 492 STATISTICAL PHYSICS [Part IV The first integral is calculated immediately; it is (44.37) We change the integration variable in the second and third integrals, assuming Then the second integral reduces to the form; — oo — oo because the integrand is an odd function. Finally, the third integral (see Appendix) is x^dx (e* + 1) (6-* + 1) (44.39) Thus, the required integral appears in the form of the following expansion: n /= y((a) - Y(0) + (g) =|Y'(e)de + (44.40) 0 The zero term in this expansion corresponds to the form that the Fermi distribution has at absolute zero; indeed, if /=! for 0)'/» /• e’/ade I ’ ./ e 0 + 1 N = F(2»l*)Va /" eVjrfs I c-tt .1 e 0 +1 (44.41) (44.42) We apply formula (44.40) and obtain W = (I +4-4F-'-.e-). F ( 2 w ’)'/2 (44.43) (44.44) because the function Y'(e) equalled e’^« for the first integral and e‘/a for the second integral. Using these formulae let us find the specific heat. From the defini¬ tion of specific heat we have SO (44.46) We calculate the derivative from the second equation, differen¬ tiating it as an implicit function: hi so dJV SO dN ■S(. 6 IX • (44.46) We have both times omitted difiFerentiating the coefficient of 0, because 6 is regarded as small. Substituting (44.46) in (44.45), we write the specific heat as S^ ( 44 . 47 ) ^ so Finally, in place of p we must substitute the expression for the limiting energy (44.7). Then the specific heat will be expressed in terms of the gas density and temperature: 494 STATISTICAT, PHYSICS [Part IV 0 Thus, the specific heat per electron is approximately 5—, which, according to (44.9), is a very small quantity. For example, we esti¬ mated that for sodium Eq-- 34,800°, so that — ~0.01 at room temper- ^0 ature. The siiecific heat of a Fermi gas per electron at room tempera¬ ture is 0.0.5. This must be comjiared with the specific heat for a Boltz¬ mann gas, equal to 1.5 from Sec. 40 (if 0 is expressed in ergs, the specific heat C is an abstract quantity). It is easy to see why the specific heat of a Fermi gas is considerably less than the specific heat of a Boltzmann gas: not all the electrons in a Fermi distribution arc capable of being thermally excited, but only those whose energy is close to the critical energy. This is why the specific heat of a Fermi gas turns out equal to a few per cent of N, 3 A specific heat -- N is obtained only when all the electrons are capable of being thermally excited. I)i!ficuUics in the classical electron theory of metals. Considerable difficulty was experienced in the prequantum theory of metals because the electron gas in a metal does not have an experimentally noticeable specific heat at room tenqierature. The specific heat of a met.al does not exceed the value 3 per atom [see (42.32)]. Yet if the number of elec¬ trons present equalled the number of atoms, then, according to classic¬ al statistics, the metal would have a specific heat 3 -j- 3/2 = 9/2 per atom, which is never observed. If we ap])ly Fermi statistics to electrons, then, as we have just seen, the difficulty with siiecific heat is removed. At low temperature the specific heat of the crystal lattice of a metal is proportional to 0® [see (42.35)]. Therefore, if the tenqieratnre is sufficiently low, the electronic specific heat begins to predominate and can be measured. Measurements show that at very low tempera¬ tures the specific heat of metals is indeed proportional to 0. As can be seen from (44.48), if we know the specific heat we can also determine the number of electrons per atom. It is a curious fact that bismuth, which in many respects is not a typical metal, has a very small number of conduction electrons. Exercises 1) Find the equilibrium concentration of electrons and positrons in some volume not containing charges at low tompcratui'e. In place of tho conservation of the number of particles we must take into accoimt the conservation of charge in the formation and aimihilation of elec¬ tron-positron pairs. Denoting the number of electrons in a given quantum state by tho letter /, and the number of positrons by the letter we have, in place of (39.23), the following supplementary condition: yjgk{lk-rk)^o. k See. 44] FEBMI DISTBIBITTION 496 Determining f and f', which give the maximum of the fimction S = ln P with the supplementary condition indicated, we obtain the distribution fimc- tions for electrons and positrons: < e - n ’ ' " e + n ’ ' 6 ® h 1 6 ® -1 1 where the constant n is the same. The total niunbor of electrons must equal the total number of positrons, i.e., \/e (/e e -t M- 1 This equation has a solution only for [i. - 0. Hence, the total number of elec irons in unit volume is p“ (Ip e 1 Let us calculate this integral when 0 <| Wo can take tho nornelati- vistic approximation for the energy and represent the distribution function in e the form c ® . Whence we have the equilibrium electron density 1 ■}UC’" This quantity is equal to 1/cm® for 0 - - 8 kov. 'The energy of tho olectro- ma^etic field per unit volume at the same tomporatines is 0.6 x 10'* ergs, while only 1.6 x 10"* erg is released in pair annihilation. 'The energy of electrons and positrons will be close to the electromagnetic field energy only when 0 is of the order of me". 2) Find the limiting energy of a superdeuse electron gas, for which tho de¬ pendence of energy upon momentum is in the main extremely relativistic: c — cp. Determine the density at which the gas may bo regarded ns ultrarola- tivistic. In place of equation (44.8), we have so that _ N_{2r.hf 3 c*“”2 ~V~’ 2Tzhc. The rest energy can be neglected if Eo > me*. 496 STATISTICAL PHYSICS [Part rv so that the condition for the density is written in the form N V 1 StcM A / «» 10®® electrons/cm®. Since Eq involves m- the inequality must bo great. The energy of suet an ultrarolativistic gas is given by the expression 3) Find the number of electrons passing through tho sui-face of a metal ill unit time if only those electrons can cross tho surface for which the velocity component normal to tho wall is greater than I'ox- This quantity satisfies the inequality In other words, the energy of the emerging electrons differs from the limiting energy by an amount considerably greater than 0 (thermionic omission). The number of electrons with velocity Vx falling on a square centimetre ol surface in one second is Vxdn{Vx), whore dn {vx) is tho density of electrons having a given value of velocity projec¬ tion Vx- Like (44.3), wo write dn (vx) in tho form dn (t'x) = 2 m® dvx dt’y dvz 1 whore e = - (I’j l- yj-|- I'J). Tho surface of a metal is crossed only by those elec¬ trons for which the difference e—g is considerably greater than 6, so that wc are justified in passing from a Fermi distribution to a distribution of the Boltz¬ mann type, but with tho same value of n as in the Fermi distribution. In othei words, wo take only tho “tail” of tho Fermi curve where e—g > 0. Whence, the ro(|uirod electron flux is OO CO oo 2 m* e 0 mi^ov^TcO mO* "“F") If we apply an electric field to the metal, the maximum current that can be extracted at a given temperature (saturation cun-ent) is determined by this fonmda. Since it relates to electrons in a metal, tho quantity g is close to go (i.o., to the limiting energy at absolute zero) and does not depend upon temper¬ ature. Sec. 44] FEftMI DISTBIBUTION' 497 It will be noticed that if we apply a very strong electric field to the metal, electrons will emerge from it overcoming the potential barrier which appears at the boundary under such conditions (cold emission). But this requires very large fields. Cold emission is analogous to the ionization of atoms in the Stark effect (seo Sec. 35). 4) CJalcidate the total energy of the electrons in an atom in accordance with the Thomas-Fermi statistical model. From (44.10), the kinetic energy of the electrons is ^0^ • 4 " 4 (6cp)‘/. dr. because the limiting kinetic energy of the electrons is eqj. We substitute instead of eiji and go to non dimensional variables (44.23). Then for we obtain OO The potential energy is divided into two parts: the interaction energy of the electrons with the nucleus, equal to ^pot — ~ J n • An dr, 0 where the electron density is determined from (44.18), and the interaction energy between the electrons themselves. 1 r Yj~(l-^)n-Anr^dr. 6 The factor takes into account that each electron should be counted once. Combining both parts of the potential energy, we have 1 r Ze^ “^pot = <^y' Substituting these integral values in the expressions for when t tends to in¬ finity (see Sec. 39). Here, t denotes the observation time for the whole closed system, which includes the given quasi-closed subsystem. There¬ fore, by its very meaning, P (S) caimot depend upon time, because this is a resultant average quantity for large intervals of time. But if P (S) is a constant quantity and g (S), as a function of the integrals of motion, is also constant, then p (S) is also independent of time and is an integral of motion. But since all the integrals of motion are, in principle, known from mechanics, p must be their function. In other words, p cannot depend upon quantities that vary with time, and, apart from S, depends only upon the integrals of motion. More exactly, p remains constant over intervals of time for which the quasi-closed subsystem may be regarded as closed. The statement concerning the constancy of p (S) is known as Liouville’s theorem. At the end of this section, a classical formulation of Liouville’s theorem will be given that is more vivid than a quantum formulation. The theorem ol multiplication of probabUities. Over a certain interval of time, quasi-closed subsystems may be regarded as independent. Then the well-known theorem of probability multipMcation can be applied: the probability that one of the subsystems is in a state A and another in a state B is equal to the product of the probabilities corre¬ sponding to states A and B. P.S = P. Pb. (46.2) The statistical weights of the states, g^ and gg, are of course multi¬ plied because they relate to different subsystems: Thus, 9AB=dA-QB- Pab ~ Pa Pb ~ ^ AB " ^AB ~ ^Ja9a' 9b 9b' (45.3) It follows from formulae (45.2) and (45.3) that Pab Pa ' Pb- (46.4) In other words, the probability density for two quasi-independent subsystems is a multiplicative function, i.e., it is obtained by multi- pl 3 dng the separate p functions. 602 8TATISTICA1, PHYSICS [Part IV Gibbs distribution. The logarithm of probabihty density is an addi¬ tive quantity, i.e., it is equal to the sum of the logarithms of this quantity for each subsystem separately: lnp^^ = lnp^-f Inp^. (46.5) We know from LiouviJlo’s theorem that in p is, in addition, an in¬ tegral of motion. Hence, In p is an additive integral of motion. In Sec. 4 of Part One we listed the additive uitegrals of motion: energy, linear momentum, and angular momentum. Por In p to be an additive integral of motion, it must depend linearly upon energy, linear momentum, and angular momentum. If we choose a reference system in which the subsystem as a whole does not move, then the linear momentum and angular momentum will be equal to zero and the logarithm of the probabihty density wiU turn out to be a linear function only of energy. In other words, the following relationship results: lnp = a^-f6. (46.6) The coefficient a must be the same for all subsystems of the large system because, otherwise. In p will not have the properties of an additive function. If a is the same for two subsystems, then these two subsystems yield 1“ P.IA == In 9a + >n 9« = «'A + ^b) + {^>a + ^b) = = + (45.7) whence the additivity of In p can be seen. The probabihty of an infinitely large energy must be infinitely smaU because a<0. We shall write (46.8) The meaning of the quantity 6 is the same as in the previous sections: it is the temperature multiplied by the Boltzmann constant. Indeed, for an ideal gas, a single molecule can be regarded as a separate sub- sjrstem, and then the Gibbs distribution of the form becomes the Boltzmaim distribution In addition, we denote (46.9) Sec. 45] GIBBS STATISTICS 603 Finally, tlie required distribution function is f--02, then d^i<0, i.e., the first system transmits energy to the second. The transmission of energy is entirely due to contact mteraction, i.e., to the microscopic forces between molecules. The energy transferred in this manner is termed heat, so that heat is not “a form of energy” but a mode of energy transmission (we shall ex¬ amine this question in more detail further on). In formula (46.4), 0^ and 02 are parameters in the Gibbs distri¬ bution for each of the subsystems separately. As long as these para¬ meters differ, the subsystems caimot occur as a unit in a state of statistical equilibrium. Approximation to equilibrium occurs as a result of heat transfer, with the heat always going to the subsystem in which the parameter 0 is least. Only then are and 02 the same, the macroscopic quantities of heat are no longer transferred, and the energy of each subsystem exhibits only small fluctuations in the vicinity of its equilibrium value. If one of the systems is an ideal Boltzmaim gas, then, as we know, 0 is proportional to the absolute temperature, since the Gibbs distribution for a gas as a whole loads to a Boltzmann distribution for the individual molecules with the same jiaramoter 0. The absolute temperature of a gas can be deter¬ mined from independent (not thermal) measurements in the Clapeyron equation 'pV — RT. It is natural to consider that the quantity 0, for any system other than an ideal gas, is also nothing other than temperature. If a system is in equilibrium with an ideal gas, then its value 0 is proportional to the absolute temiierature of the gas. Thus, 0 is the temperature measured in absolute units (ergs) if the ideal gas is taken as a thermometric substance. A little later in this section a definition of temperature will be given which does not depend upon the choice of the thermoraetric substance. A Gibbs distribution occurs for any group of quasi-independent subsystems, including those that have not arrived at a state of mutual statistical equilibrium. Even though the quantity 0 in this case, too, is, by definition, the same for all subsystems—which follows from the multiplicativity of the distribution function p (S) [see (46.4)-(45.8)]—it must not be regarded as equal to the temperature of the large system, which, generally speaking, cannot be defined for a system not in equilibrium. If the subsystems in this case are in internal equilibrium, they are characterized by their ovm Gibbs Sec. 46] THEBMODYNAMIC QUANTITIES 615 distribution, which cannot be represented by a factor involved in the Gibbs distribution of the large system because the parameters 6 of both distributions are different. Thus, the distribution modulus 6 of an equilibrium system is a measure of its temperature. Taking the example of temperature, it can be seen that quantities which are defined statistically can be identified with actually measured thermodynamic quantities. Any statistical quantity can be regarded as defined when, and only when, there is given a unique group of operations (of measurement and calculation) relating this quantity to real macroscopic quantities or to the microscopic parameters of a system which are found (or can be found) from experiment. Work. The Hamiltonian function of a system usually depends not only upon generalized coordinates and momenta that vary according to dynamical laws, but also upon certain arbitrarily chosen parameters. The intensity of the external field, for example, may be such a parameter. The energy spectrum of the system, and hence the mean energy S' also, depends upon the parameters appearing in the Hamiltonian. These parameters, transformed according to a given law, are termed the external parameters of the system. We denote them by the letter X, where X may mean any quantity of this tyi)e. As X varies, the moan energy of the system also varies. It is obvious that it can only vary at the expense of some external source of energy. Since X is a mechanical and not statistical quantity (X is involved in the Hamiltonian!), the variation in X is due to certain external mechanical work performed on the system, for example, a falling weight or a rotating motor. The mechanical work performed with changing X can be represented as dA=Ad\, (46.5) where it is natural to call the quantity A the generalized force (since work is equal to the product of “force” A and “distance” d X). If the entire energy change is due only to change in the external para¬ meter X, then dS==~d-k. (46.6) In formula (46.6), dA is the work performed on an external object due to a decrease in the energy of the system, so that dA — — d S. Comparing (46.6) and (46.6), we see that the mean quantity is equal to the generalized force taken with opposite sign; 516 STATISTICAI, PHYSICS [Part IV The most frequent external parameter of a system is the volume that it occupies. In mechanical terms, this may be visualized by considering that the potential energy of any particle included in the given system is equal to infinity beyond the boundaries of the volume, i.e., an infinite amount of work is required even to remove a single particle from the volume. This is how the volume appears in the Hamiltonian of a system. Wo shall consider that a system occupies the volume of some cylin¬ der with a movable piston. The force acting on unit area of the piston is called the pressure and is denoted by the letter p. Then, if the whole area of the piston is /, the force acting on it is pf. When the piston is displaced through a distance dx, a quantity of work dA— pf (lx is performed on it. But the product fdx is equal to the vol¬ ume increment dV of the system. Hence, the change in energy for the system is dS=—pdV (46.8) This type of energy change, produced by a change in the external parameters, is called work performed on a system. It can bo seen from formula (46.8) that pressure is a generalized force A rebated to a volume increment dV. The first law of thermodynamics. It has already been pointed out that energy c. = QdS-dA, (46.16) whence it follows that _ %d8 = dS + dA. (46.17) But the right-hand side of the last equation is nothi^ other than the quantity of heat d Q received by the system. Hence, in a reversible process dQ^^dS. (46.18) This is one of the most important equations in thermodynamics. It determines the entropy increment of a system in terms of the quantity of heat directly measured from experiment. It is significant that the quantity of heat obtained by a system in a process depends upon the development of the process, while the entropy increment is determined only by the initial and final states of the system. The quotient ob¬ tained from the division of an infinitely small quantity of heat (received by the subsystem in a reversible process) by the temperature is the total differential = d8. (46.19) If an irreversible process occurs inside the system, equation (46.19) may not hold. Indeed, let the system consist of two subsystems at different temperatures. In the process of temperature equalization, such a system approaches statistical equilibrium and its entropy increases. But no heat reaches the system from outside, so that dQ for the whole system is equal to zero and d8>Q. Let us consider another example of an irreversible process. Let a gas expand into a vacuum. The phase volume AT [see (45.35), (45.39)] Sec. 46] THERMODYNAMIC QUANTITIES 621 naturally increases, since the geometrical volume increases. But this means that the entropy also increases. When expanding into a vacuum, the gas does not perform work (since there are no opposition forces) and does not receive heat. In other words, it may be regarded as a closed system whose entropy increases as statistical equilibrium is approached (when a gas expands isothermally in a cylinder situated in an external medium, the entropy of the gas also increases, but the entropy of the medium decreases to the same extent). Consequently, the entropy increment in the case of an irreversible expansion of a gas into a vacuum is positive, and the quantity of heat transferred is equal to zero. The two foregoing examples show that if an irreversible process occurs inside a system, then ^0. Reversing the sign of the inequality, we get -d(?-0/S). (46.34b) The quantity I-QSsF (46.35) [cf. (45.31)]’appears in the Gibbs distribution (45.10); it is called the free energy of the system. It follows from the inequality (46.34b) that the greatest amount of work that can be performed by a system at constant temperature is equal to the change in F, taken with opposite sign: = (46.36) Thus the work is equal to its maximum value in a reversible process. The inequality (46.34a) has a somewhat different meaning: it deter¬ mines the least amount of work which must be performed on the system in order to produce the given change of state in it: Sec. 46] THERMODYNAMIC QUANTITIES 625 = (46.37) The entropy of the system and the surrounding medium (taken together) is conserved in these processes, and the inequality (46.32) becomes an equality. Consider the following example. Let an ideal gas expand into a vacuum. No work is performed so that energy is conserved. But the energy of an ideal gas depends only upon its temperature (see Sec. 40), and not upon volume. Therefore, the temperature does not change during expansion into a vacuum. As we have seen, the entropy of the gas increases. Then the minimum work required to return the gas to its original volume at the same temperature is equal to the change in its free energy during expansion. The entropy of the gas will de¬ crease in such a reversible compression, but on the other hand the entropy of the surrounding medium will increase to the same extent. It is easy to obtain a thermodynamic identity for free energy. Differentiating the relationship between total and free energy, and substituting the identity (46.26), we obtain dF = - SdQ - pdV, (46.38) whence it is easy to find an expression for the entropy and pressure, and also an equation between the cross derivatives The relations (46.39) are convenient in that the independent variables are volume and temperature, which can be directly measured experi¬ mentally. Yet the thermodynamic identity for energy (46.26) involves entropy as an independent variable. But the entropy itself must be calculated, for example, by integration of (46.19). From (45.12), the free energy F is expressed in terms of a statistical sum _ ^ F = -ein2^c 9. (46.40) The right-hand side of this equation is expressed in terms of the temperature 0 and the external parameters involved in the character¬ istic values of But 0 and X are those very independent variables which are chosen in the identity (46.38). Therefore, for a determination of all the thermodynamic quantities it is sufficient to calculate the _ £ statistical sum^e 9 . The actual calculation of this sum for an arbitrary system entails enormous mathematical difficulties. Actually, it is calculated only for ideal gases and crystals, and also for systems which deviate but little from ideal. It should be noted that even if it were possible to evaluate the statistical sum for some actual substance. 626 SXATISTIOAIi PHYSIOS [Part IV say liquid water, the thermodynamic laws obtained with such very great difficulty would apply only to water and not to liquids generally. But the properties of ideal gases and crystals follow from statistics in a very general way. Thermodynamic potential. Let us now determine the maximum work that can be performed by a system at constant temperature and pressure, equal to the temperature and pressure of the external me¬ dium. Wo note that m a homogeneous system with a constant number of particles, where there are no phase or chemical transitions, the state is completely defined by the temperature and pressure, since the thermodynamic identities for such systems involve two independent variables. In this case the .specification of two quantities determines all the rest. But if a system consists of two phases of the same sub¬ stance, for example, a liquid and its vai)our, then the relationship between the fractions of liquid and gaseous substances may be quite arbitrary for a given temperature and pressure. Work is performed in increasing the volume of a system. Wo can imagine, for example, a system in a cylinder under a piston, and the piston rod connected with some object capable of changing only its mechanical energy: by means of a flywheel or load. In addition, on expansion of the system work is done on the external medium. If we call the work on the object A, then the total work performed is equal to — A—pA V= — (A-\-ApV). Here, p is the pressure in the external mechum, which pressure in the process considered is equal to the pressure in the system. Since, by convention, the temperature of the system does not change, we have, from (46.33), - {A + Apr)^A{^-QS) or — A^AiJ—OS + pV). (46.41) The quantity tu —is, obviously, a function of the state of the system. It is called the thermodynamic potential and is denoted by the letter : (46.42) Thus, the maximum work which a system can perform at constant temperature and pressure is equal to the change in its thermodynamic potential (with reversed sign) •^max “ (46.43) This work is performed in a reversible process. When equilibrium is established in a system, work cannot be performed. Then d) attains a minimum, because, according to (46.43), the work is performed at the expense of a decrease in <&. When Sec. 46] THEBMODYNAMIO QUANTITIBS 527 vlmax = 0, O cannot decrease further. It has already been pointed out that the process can occur at constant temperature and pressure with a phase transition or chemical transformation; hence, the equilibrium condition here is that should be minimum. Let us now find the thermodynamic relationships for ^>. From (46.42) <5 = F + •pV. (46.44) Differentiating this equation and substituting dF from (46.38), wo obtain d dp 89 * (46.46) The thermodynamic potential depends only uijon quantities that characterize the state of a body: its temperature and pressure. At the same time 4) is, of course, an additive quantity; if two equal volumes of the same substance are joined at the same temperature and pressure, the common thermodynamic potential will be twice as great as it was for each part separately. Therefore, we can write O = A (i. (p, 0). (46.47) Here, (a is the thermodynamic potential related to a single molecule of the substance, p is also called the chemical potential. We shall show later on that for ideal gases it is identical to the parameter (a in the distribution fmiction (see Sec. 39). It is obvious that (46.48) If the system consists of several types of molecule, for example, a solution of one substance in another or a mixture of gases, then the state is determined not only by the temperature and pressure, but also by the concentrations of the substances. The concentration of the I’th substance in a mixture is (46.40, The chemical potential of the ith substance in a mixture is natur¬ ally expressed by analogy with (46.48): _ / 80 \ (46.60) 628 STATISTICAL PHYSICS [Part IV where (a,- depends upon p, 0 and all the concentrations; c^, a, - Regarding Ni as variables, we can write the total differential of d> in the following way; d(^^-SdQ + Vdp+2Jv^dNi, (46.51) i This equation generalizes (46.45) for the case of a variable number of particles. Since the transition from S to F and d> does not involve the number of particles Nu we can similarly generalize the differential relations (46.26) and (46.38): de = ^dS-pdV + 2^^,dNi, (46.52) djP = - 6 - pd F + 27F>- dNi. (46.53) For a constant volume and for one type of molecule, (46.52) reduces to the form dS = QdS + y.dN. (46.54) But this equation coincides with (39.18), whence it can be seen that the quantity S introduced in Sec. 39 is the entropy of a gas and [a is its chemical potential. Entropy in classical and quantum statistics. Let us compare the definition of entropy based on classical and on quantum laws of motion. In the latter case, entropy is defined as the logarithm of the number of states of a system for a certain energy value. When passing to a quasi-classical approximation, the number of states of the system is equal to the phase volume AT it occupies divided by (2 Tt h)", where n is the number of degrees of freedom [see (45.39)]. The logarithm of this ratio represents the entropy in the corresponding approximation. But statistics appeared before quantum mechanics. Therefore, entropy was originally defined in statistics as the logarithm of the denominate number A F. In this definition, entropy depends upon the choice of units: if, for example, the unit of mass is ehanged by a factor two, then nln 2 must be added to the entropy. But since the units of measurement are arbitrary, it follows from this that in classical statistics entropy was defined only within the accuracy of an arbitrary additive constant. Only the change of entropy had strict meaning. In quantum statistics, entropy is defined as the logarithm of an abstract number, and therefore does not depend upon the choice of units of measurement. Sec. 46] THERMODYNAMIC QUANTITIES 629 The temperature of a system is equal to absolute zero when the system is in the ground state, i.e., when it has the least possible energy. This state has a weight equal to unity, so that the entropy, or log¬ arithm of the weight, becomes zero at the absolute zero of temperature. This statement is known as Nernst’s theorem, which is sometimes called the third law of thermodynamics. Certain consequences of Nernst’s theorem will be considered below (see exercise 6). Exorcises 1) Find the ratio between the specific heats at constant volume and at constant pressure. With the aid of (46.18), we find, from the definition of specific heat. Tile derivatives can be rewritten in the following way: from the formulae for the derivatives of implicit functions. The partial deriv¬ atives with the same subscript may be cancelled like fractions, since the differentials in them have the same sense. T'his gives I8p\ 1 \8V}s^ [dp to cv (^] \8V}o \dpls so that sj)eciflc heats are related in the way that isothermal compressibility relates to isentropic comiiressibility. It is sufficient to measiue only three — ,ev\ of the fotir quantities Cp,Cy, \Wp}s ( Q ^ \ 8V to and, according to (46.39), transform the We substitute S —F+tiS-, derivatives ~ IdF as dVh = — p+0 . If the pressure is known as a function of temperature and volume, the energy can be calculated only to the acctiracy of an arbitrary temperature function ?.Jdr[-p+e(lf)J + /(o). Therefore, it must always be remembered that a determination of the relation¬ ship p =p(F, 0) does not yield complete information about the thermo¬ dynamic properties of a substance. In addition, any pressure term depending 34 - 0060 630 STATISTIOAl PHYSICS [Part IV linearly upon temperature will not affect the energy, since it is eliminated N 0 from the equation obtained. For example, in all ideal gases p = —pr- , and the energy depends upon the temperature in a rather complicated way if discrete quantum levels must bo taken in the statistical summations. 3) Answer: V + 4) Find the difference between the specific heats at constant volume and at constant pressure. The quantity of heat at constant pressure is equal to dl, and at constant volume, to dS [see (40.14) and (40.12)]: We transform Cp: Further, representing energy as = # [0, F {p, 0)], we write the deriv'ative (■^-] m the form \S0 /p _ _ _ _ 00 Whence where we have used the result of exercise 2. The derivative thus: ) is transformed e/p ( 8Q I u Whence it follows that Cv = (IP) \ 80 Ir \8v/e Aso/y (iP) [ev/e It will later bo shown rigorously that|-^^|^< 0, i.e., the pressiue can only increase with decrease in volume (otherwise the state of the system is dynam¬ ically unstable, which is obvious as it is). Therefore Cp > Cv always, and also I8p\ _ N (IP) _ NO \ SO jy V • \eF/e y2 so that Cp — Cy = K. 6) Accepting the second law of tlierraodynamios as a postulate, prove that the efficiency of a reversible engine is always greater than the efficiency for an irreversible engine, working with the same temperatiue difference between sowce and sink. Sec. 46] THERMODYTTAMIO QUANTITIES 631 The proof is indirect. Let a reversible engine and an irreversible engine obtain the same quantity of heat Qi from a source, but let the in-evorsible engine give a smaller quantity of heat Q^' to the sink than the reversible engine. The reversible engine may be made to work as a refrigerator, i.e., to take heat from a cold reservoir and to deliver it to a hot reservoir at the expense of external work. In order to return a quantity of heat to the hot reservoir, in accordance with om’ assumption, the reversible engine must take a larger quantity of heat from the cold reservoir than the irreversible engine delivered. But it will then turn out that the hot reservoir does not deliver heat at all, and the cold reservoir delivers a quantity of heat — Q./, at the expense of which useful work is performed equal to the difference between the work of the irreversible engine and the work of the reversible engine operating as a refrigerator. The surrounding medium can be taken as the cold reservoir, so that usofid work will be performed at the expense of heat obtained from the surrounding medium; this contradicts the second law of thermod 3 mamics. 6) Prove that the specific heat of a system tends to zero when the tem¬ perature tends to absolute zero. Do the same for The entropy is related to tho specific heat O by the relation 0 i^v\ _ (SS\ \ 80 )p Up/' where the lower limit of the integral is put equal to zero from Nernst’s theorem. For the integral to have meaning, we must demand that lim O = 0. In addition iSS\ . — limit of tho last derivative is also equal to zero as 0 tends to zero, because lim iS = 0 in the case of an arbitrary pressure. o-».e 7) Show that the sum of the enthalpy and kinetic energy is conserved in the motion of a substance without any internal heat exchange and without exchange of heat with tho external medium. Let a certain mass of substance be transferred from a volume Vi, pressure Pi, and energy to Fj, p^, and respectively. In order to displace this mass from the volume Fj at a pressure Pi, an amount of work pi Fj must be done. Therefore, in going to p„ V^, a work p^ Fj — pj Fj is done. The total change of energy of the given mass, in a reference system fixed in it, is equal to d’j — + Pi Fi — pj Fj = Ii — Jj. Since there is no heat exchange, this quantity can be equal only to tho change in kinetic energy mvl mvl — 7 7 “2““ ~ ”2“ mv\ mv} 2 + In future we shall relate this equation to imit mass of the substance, and write it in the form: ti* I -f =» const, where I is the enthalpy of unit mass. 34 ' 532 STATISTICAX PHYSICS [Part IV 8) Find tho propagation \'olo(:ity of small isontropic tlistnrbances in an isotropic medium [in other words, neglecting heat transmission and considering that p == p (p)]. If the initial position of a particle is described by a single coordinate a, and tho displaced position by tho coordinate x, measured in the same direction, then the equation of conservation of mass is the following; Po da — p dx (p is the density, p,, is the initial tlonsity). Wlience Po \9»/(’ Tho force acting on an element of mass p dx is — p (a; + dx) + p (;c) = — dx. According to Newton’s Second Law, this force is equal to the product of mass and acceleration, i.o.. Considering tho displacements small, wo see that the derivative ^ is close to imity, so that the second derivative is a small (piantity. The result, therefore, is the approximate equation 8‘x 1 _ fi 3a® (^ {wls It coincides with tho wave equation of the form (17.4), which describes the propagation of ilisturbanccs with velocity c. In the given case, the propagation velocity of the process is nothing other than tho velocity of sound. It is equal 9) A substance flows in a tube of constant cross-section without heat exchange, but with internal friction. Show that the maximum entropy is attained where the flow rate is equal to the velocity of sound. The following conservation laws apply: pv — j = const. 1 + 2 = const. Sec. 46] THERMODYNAMIC QUANTITIES 633 Substituting the velocity from the first equation, wo obtain Let us differentiate this equation, considering that the enthalphy is expressed in terms of the independent variables S anil p: P* d p = 0. Close to tho entropy maximum dS = 0, the derivative so that V P ( (>p jdS^-0 For constant entropy so that at maximum entropy. If there is a flux in the tube for which v < u (“subsonic flow”), the value V — u can only bo attained at the tube outlet because, otherwise, tho entropy, on reaching a maximum somewhere in tho tube, would have to decrease in the subsequent flow, which is imjiossible. 10) A substance flows without heat exchange or friction, i.e., isenlropically, in a tube of continuously variable cross-section /. Show that tho velocity in subsonic flow increases with decrease in cross-section, but in suporsonie flow, it increases with /. The flow is considered as one-dhnensional because of the smooth variation of /. From tho law of conservation of mass whence it follows that / p a = const, df ^ ^ - 0 f p V ' Taking into account that entropy is constant, wo can write: P P I If.) = dp Is p Differentiating tho equation I -)- = const at constant entropy, we have dp P -f vdv = 0 Wlicnco it follows that d p vdv and finally P dv / v^ \ df ~v' f which proves the statement. In order to obtain flow with supersonic velocity at the outlet of the tube, we must pass it through a Laval nozzle, i.e., along a tube whoso aperture first decreases, so that v — u at the narrowest place, after which v becomes greater than u and continues to increase. 634 STATISTICAL PHYSIOS [Part IV 11) A piston is in movement with constant velocity v, into a cylinder with cross-section /, filled with a substance with initial pressure Po initial density Po. The enthalpy I per vmit mass is regarded as a known fimction of p and p. Formulate a system of equations, from which it is possible to determine the displacement velocity of the boundary between the compressed and non- compre&sed substance, and also the density and pressure of the compressed substance. The compressed substance moves with velocity v equal to the piston veloc¬ ity. The boimdary between the compressed and noncompressed substance has a certain velocity D. The compressed substance has a velocity v — D relative to this boundary, and the noncompressed substance has velocity D. Let us pass to a reference system moving together with the interface. Then the mass conservation law is expressed as follows: /PoD-/p(D-e). (•) We shall consider a cylindrical volume of the substance passing through the boundary in unit time. The length of this cylinder in the compressed sub¬ stance is equal to Z? — v, while its mass is f f (D — v), so that its momentum is equal to f p (D — «)*. The momentum in the noncompressed substance equalled / po'Z?®- A resultant force (po— p) / acted on this cylinder, whence the conservation equation /(Po+Po-D’) = /[P + P(J5-«)*]• (**) The third equation expresses the absence of heat exchange (see e.xercise 7) It) -h 2 1 + (D - e)» 2 (*•*) At the interface, a discontinuotis change occurs in the density, pressure, and velocity of the substance. This surface is called a shock wave. We can deter¬ mine D, p, and p from the three conservation laws, if the form of the function 1 (p, p) is known. These quantities will be determined specifically in exer¬ cise 7, Sec. 47, where it will also be shown that the compression process in a shock wave is irreversible. 12) Show that the classical expression for a statistical sum does not depend upon the constant magnetic field in which the system is situated. The classical expression for the statistical sum (or more exactly, for the inte¬ gral) is .Pjv;r»>r,,.. . r^) Z = J e ® dTp,dTp,dTp,.. dVjdFj... dFN- When the system is placed in a magnetic field the momenta of the particles change according to the formula p->-p-^A=«P. Passing (for the phase- volume element) from dtp to dtp, wo find that the statistical sum appears the same as in the absence of field because the new notation for the integration variable does not change anything: _ . '«• . Z = j e ® drp, dtp, ... dtpj^j • dFj dF, ... dFiv. Thus, classical mechanics cannot describe the magnetic properties of a sub¬ stance. Sec. 47] THE THBBMODYNAMIO PBOPBBTIE8 OB IDBAI. GASES 536 13) Express the entropy of an ideal gas in terms of the occupation nk for all three statistics (gk = 1 everywhere). Using the expressions for S in equations (39.14) and (39.25), we find: Bose statistics: S + 1) In {nk + 1) - nu In njb]; k Fermi statistics: — ^[(1— nfe) ln(l— wa) + Wfclnwfc]; k Boltzmann statistics corresponding to n* 1: S=-2’n.ln^-. ;r If the weight is not equal to unity, then, introducing Hk ^ gk fkt we obtain for all three statistics: ^Boae =2^ [{/fc + 1) In (/fe + 1) -/fc In/fe] , k Si'ermi = - 2^ gk[(l — - fk) + fk In fk], k 'S'BoItemanu — 27 ^ ^ ' It Sec. 47. The Thermodynamic Properties of Ideal Gases in Boltzmann Statistics In this section we shall consider certain consequences that foUow from the general principles of thermodynamics as applied to ideal gases. We shall suppose that the gas density is sufficiently small for Boltzmann statistics to be applied to its molecules. This does not mean that the motion of the molecules should be regarded as nonquantum; the quantization of rotational, vibrational (and aU the more so, electronic) levels of a molecule must be taken into account in all cases when the spacing between neighbouring levels is comparable with 6 (i.e., kT) or greater than 0. Even when the level spacing is sufficiently small compared with 6, as is the ease of trans¬ lational motion, the quantum of action should be left in the formula for the statistical weight of the states, since it would be impossible otherwise to obtain a unique expression for entropy. Deviations from Boltzmann statistics that occur in gases at low temperatures or high densities are sometimes called “degeneracies.” One should diflerentiate between deviations from the characteristic 636 STATISTICAI. PHYSIOS [Part rv ideal gas state, due to the interaction between molecules, and quantum deviations from classical statistics. Of course, there also arise correc¬ tions which are due to the effect of both factors together. Free energy of an ideal gas. As was indicated in the preceding section, it is convenient, when calculating thermodynamic quantities, to proceed from the expression for free energy. We shall start with formula (46.40), reducing the statistical sum to the form that it takes for a Boltzmann gas. For this it is necessary to take into account that, by definition, a statistical sum is calculated over all the physically different states of a gas. But the state of the gas docs not change if all possible molecular permutations are per¬ formed over the individual states; in nonquantum statistics such a permutation can be defined. The number of permutations of N molecules is equal to A!. The total energy of an ideal gas separates into the sum of the energies of all of its molecules: N ;=i where k is the number of the quantum state. Substituting the expression for into the statistical sum (46.40), and dividing this sum by the number of permutations of the mole¬ cules, we obtain N , -ly' 9 * 1-1 N fc Vfc_ Nl Nl Nl (47.1) The second summation over (k) relates to all possible combinations of the energy of the separate molecules Here, we have made use of the fact that the energy spectrum is the same for all molecules (if the gas consists of molecules of one type). The summation in (47.1) is performed over the spectrum of a single molecule. Replacing A! by its expression in Stirling’s formula, we arrive at a general formula for the free energy of an ideal gas under Boltzmann statistics. e(k) 6 F = -NQ]n-^ -. (47.2) Summation over translational degrees of freedom. It is expedient, in the statistical summation over the states of a separate molecule, to separate the translational degrees of freedom and represent the energy in the form Sec. 47] THE THERMODYNAMIC PROPERTIES OF IDEAL OASES 637 e = + sC). (47.3) It is taken here that the energy does not depend upon the coordinates of the centre of mass of the molecule. The statistical weight of a state with momentum p is equal to g = g^'> dpx dpy dpz dx dy dz (47.4) Sf('> denotes the weight referring to an energy level sW. Integration over X, y, z contributes the factor J dx, dy, dz=V to the statistical sum. The integration over momenta is performed in a familiar manner: Je dpx=V'lT:m^. (47.6) Thus, the free energy for an ideal gas reduces to the following form : i’= _iV01n^/(0). (47.6) A relationship is obtained here between free energy and volume. The function / (0) depends upon the molecular structure. Thermodynamic quantities of an ideal gas. It is easy to determine pressure from formula (47.6). From (46.39) we obtain dF NQ 'P~ dV ^ V ’ (47.7) i.e., the well-known Clapeyron equation. The thermodynamic potential is O = F + pF = F + iV0 = - A61n^/(0), but here it is expressed in terms of volume. To be able to use identity (46.45) we must, in addition, replace ^ by , whence a final formula is obtained for the thermodynamic potential of an ideal gas: a) = -A61n-^^. (47.8) We find the chemical potential with the aid of (46.47) or (46.48): (j.= -01n-^^. (47.9) The entropy of an ideal gas is = Wln^/(6)+Ae rm /( 0 ) ' (47.10) 538 STATISTIOAl PHVSICS [Part IV This expression does not agree with Nernst’s theorem. In actual fact, of course, we must apply to a gas at very low temperatures, not Boltzmann statistics but quantum statistics, even neglecting the fact that at low temperatures the gas actually condenses. The energy is equal to S—F + 6 t us bring together the laws from which the specific heat of a gas subject to the equipartition principle is calculated. At a sufficiently N high temperature there is a specific heat -j- per rotational degree of freedom, and also per translational degree of freedom, since each such degree of freedom contributes one squared term to the energy expression of the form • Hence, the integral of the distribution function acquires either a factor V27t m 0 or V27t ; this yields the term when calculating the energy. If we can replace the summation by an integral, the vibrational degree of freedom contains two variables appearing quadratically in the energy [see (47.22)], thus jdelding the mean energy 0. To summarize, at a sufficiently high temperature, each vibrational degree of freedom, if it is strongly excited, makes a contribution N to the specific heat. If we apply the equipartition principle, then the specific heat of a molecule consisting of i atoms, which are not in Sec. 47] THE THERMODYNAMIC PROPERTIES OF IDEAL GASES 543 one line (i> 2 ), is equal to —~j N, and if the atoms form a line in the equilibrium position, it i-s (3 i — 3) N. Thus, for a triatomic molecule of triangular form (for example H 2 O), a specific heat 6 iV' is obtained for full excitation of all the degrees of freedom (besides electronic), and the ratio C7 p/C'f = 7/6. If the vi¬ brations arc not yet excited, then (7 f==3 N and CpjCv—^l^. At the lowest temperature, only the translational degrees of freedom remain, as in the case of a monatomic gas, which gives Cv = ^N and Op/Cr=5/3. If the atoms of a triatomic molecule form a line (for example CO 3 ), then the maximum specific heat Cv~ — N and (7p/C'i'= 15/13, i.e., Cv is greater for a linear molecule than for a triangular molecule. But if vibrations are not excited then Cv=-^N, which is now less than for a triangular molecule. Such an intersection of the specific heat curves of COj and H.,0 witli change of temperature is actually observed. Adiabatic demagnetization. Of great interest is the process of isentropic (adiabatic) demagnetization. In Sec. 40 we considered the paramagnetism of the salts of rare-earth elements due to the free rotation of the magnetic moments of unfilled shells. Such moments may be interpreted as a “gas.” Let us suppose that a salt is magnetized to saturation at low temper¬ ature and is then suddenly demagnetized. Its entropy does not have time to change. But if all the moments are orientated in one direction, the entropy is small because this state is obtained in a small number of ways (in one way, in the limit). When the field is rapidly removed, the entropy will remain small only due to a big drop in the temperature. This method has been used to obtain temperatures of several thou¬ sandths of a degree above absolute zero. Exercises 1) Find the work and the quantity of heat obtained by a gas in an isothermal process. The work is equal to the change in free energy: A = -NQin-^. The quantity of heat is expressed in terms of entropy change: A and Q are equal and opposite in sign, because energy remains unchanged at constant temperature. 2) Two portions of different gases occurring at the same temperature and pressure are mixed. Find the increase in entropy. 644 STATISTICAL PHYSICS [Part rV A S = InIn _ JV In _ ivr In , * p, ^ Pi P P whore Pi and pj are tho partial pressures of both gases after mixing. Wlionce A S - Jr, to A + W. In i . In + », In . If two portions of tho same gas aro mixed imder the same conditions, the entropy will equal (N^ + N^) In , after mixing, so that A S' = 0 as it should be. This would not have occurred if tho factor N\ in the statistical sum had not been introduced into the expression for free energy. Due to this factor, only tho summation over physically different states of a gas appears in the free energy, and the entropy cannot change when two portions of the same gas are combined at the same temperature and at equal pressure. 3) Calculate the free energy of a gas in a centrifuge, of radius R and length I, rotating with angular velocity oj. Find the mean square distance of the particle from tho axis. The centrifugal force is equal to r, which corresponds to an effective potential energy U = — j ma^rdr — - ^ Whence we obtain an expression for the free energy R mo)*r« e/{0) F= -WOln N I e 20 r dr = — .N 0 In efM N nioi- -i) Tho free energy satisfies tho general relation d F = — S d 0 — AdX [cf. (4().38), where X - F]. Regarding as an external parameter X, we determine the mean square distance of the particle from tho axis: Nm S(to*) because, if —t simply Sec. 48] rLtrCTUATIONS 547 denotes the simultaneoua transition which can be directly- seen from Schrbdinger’s equation (24.11). But the function is completely equivalent to ij/ (no matter which of them is regarded as conjugate). In more complicated cases, when the operator ^ is complex, we can likewise always pass from the function t}* to another (physically fully equivalent to it) together with the transition from 5 to — Statistics and reversibility in time. We shall now examine the way that the laws of statistical mechanics relate to time inversion. Statistical mechanics states that if at some initial instant of time a system is deviated from statistical equilibrium, then, in the over¬ whelming majority of cases, it will subsequently approach equilibrium. A system which is already in equilibrium will remain in equilibrium, no matter what imaginable changes in the sign of time are performed in the mechanical equations describing the detailed microscopic state of the system. Therefore, a situation arises that is rather paradoxical at first sight: statistical laws, which appear noninvariant with respect to time inversion, are derived from the equations of mechanics! The problem stated in classical mechanics and statistics. Let us examine this paradox in the limits of the classical laws of motion. First take the following example. Let a gas occupy one half of a vessel divided by a partition. After this partition is removed the gas will occupy the whole vessel. Let us foUow the motion of each molecule of the gas in this irreversible process (in classical mechanics this is, in principle, possible). The motion of aU the molecules is represented in phase space by the displacement of a single point along a phase trajectory. If in the state of statistical equilibrium we mentally change the signs of all the velocities, the phase point in its imaginary move¬ ment will be displaced in the reverse direction, and all the gas will collect in one half of the vessel. Since any equilibrium state of the gas is attainable from a nonequilibrium state, and both velocity signs are, a priori, equiprobable, the gas must come out of the statistical equHibrium state as often as it enters it—^which, it would appear, is never observed. In actual fact, in statistics, equilibrium is not just any strictly defined state, but a whole range of states in which a closed system spends the greater part of its time. The phase point roams about in the equilibrium range for an extremely long time before spontaneously leaving it for any considerable distance. Through the vast majority of phase points in the statistical equilibrium region there pass trajec¬ tories which almost never enter regions that correspond noticeably to nonequilibrium states. If we choose a certain section of the equilibrium range, we may say that the system emerges from it just as frequently as it returns to it, but that in the vast majority of cases it does not go “far.” 548 STATISTICAL PHYSICS [Part IV Therefore, the apparent irreversibility of statistics is due to the way the problem is stated in it: the system does not remain for long in nonequihbrium states, and therefore rapidly enters equilibrium states; it remains for a very long time in equilibrium states, so that the probability of spontaneously leaving these states ean, in the majority of cases, be neglected. Quantum mechanics and the irreversibility ol transitions. The principle of detailed balance (see See. 39) is fundamental to quantum statistics. In accordance with this principle, the probabilities of direct and inverse transitions are equal between two states having the same statistical weight. However, it by no means follows from this prin¬ ciple that the probability of transition from an equilibrium to a non¬ equilibrium state is the same as for a transition from a nonequilibrium to an equilibrium state. A statistically equilibrium state includes very many equiprobable microstates, while a nonequilibrium state con¬ tains a comparatively small number of microstates: the reason why the system spends the greater part of its time in equibbrium is that there are incomparably more equilibrium states than nonequilibrium states. Each given microstate, belonging to the set of statistical equilib¬ rium states, passes to another state from the same range with over¬ whelming probability, while to a nonequilibrium state it passes with a negligible probability. A nonequilibrium state passes preferentially to an equilibrium state because the transition to a state of less equilib¬ rium can occur in an incomparably smaller number of equally prob¬ able ways. It is for this reason that a system “tends,” as it were, to equilibrium, despite the identical probabihty for direct and inverse transitions between any two initially equiprobable microstates. Poisson’s formula. The spontaneous transition of a system from an equilibrium state to a noticeably nonequilibrium state is of very small probability, but is not completely impossible. Deviations of actual values from their averages are more probable, the smaller the system in which they occur. If, for example, gas molecules are ob¬ served in a cube with a side 10~® cm, then, under normal conditions (0® C, 760 mm Hg) the mean number of molecules is in all 27. The molecules may leave for neighbouring portions, so that their actual number in a certain volume will exhibit a very noticeable deviation from the number 27. It is very easy to determine the probability that there will be N molecules in a given volume F, if there are Ng molecules contained in the total volume Fq. The probability of finding a single molecule in the volume F is obviously equal to . Therefore, the probability of finding N molecules in the volume F, and Nq—N molecules in the remaining portion of the volume, is equal to N„l (y\^(i V\N,-N (N,-N)lNl\Vol \ Vo) (48.1) Sec. 48] FLUCTUATIONS 549 An analogous formula was derived in Sec. 39 for the probability of obtaining tails k times. Let the total number of molecules be arbitrarily large and let N be any number, though considerably less than Nq. We replace the factorial ratio thus: ( y \N / Y N -^1 and II- y-\ as 0 0 /_7 \N_ ^ \Vol “ N,N’ — y — where A = by definition of N. Substituting all the obtained ex¬ pressions into the initial formula, we find the required probability «-N. iN (48.2) (Poisson’s formula). It will be shown in exercise 1 that at large N the distribution (48.2) has a very sharp maximum at N—N. Fluctuation probability. Here, we shall obtain a general formula for fluctuation probability in a subsystem of a large system. The small volume of gas just considered may be taken as a special case of such a subsystem. Let it be that a certain deviation from statistical equilibrium has occurred in the subsystem. The entire large system thus have deviated somewhat from equilibrium. The ratio of the probabilities for the equflibrium and nonequilibrium states of the large system is equal to the ratio of the statistical weights of the states w Q iVq Oq (48.3) where w and O refer to the large system. The index 0 denotes the equilibrium value. Expressing the statistical weight in terms of entropy (/S=ln(?), we obtain — . (48.4) W0 Formula (48.4) can be given a somewhat different form. Since the large system is closed, its energy remains unchanged for fluctuation 660 STATISTICAI/ PHYSICS [Part IV in the subsystem: But the totel energy and the free energy F are related thus: F = £ —0/S, F^^—S^ —O/Sq. It follows from these equations that the change in entropy of a system undergoing fluc¬ tuation is equal to the change in free energy, taken with opposite sign, divided by the temperature: ,8 —/S„ = ^i^=_ii^. (48.5) The change in free energy is expressed, in accordance with (46.37), in terms of the minimum work. Here, Amin is the minimum external work which must be performed on the system in order to produce this fluctuation reversibly, i.e., without change in entropy. Thus, the fluctuation probability is defined by the following formula derived by Einstein: W'^ e ® . (48.6) It must be borne in mind that fluctuation occurs spontaneously, with¬ out expending any external work. This was taken into account by the equation The same deviation of actual values from equilibrium values in the subsystem can be produced without expecting a fluctuation in it—^by performing work Amin reversibly. Fluctuations ol thermodynamic quantities. The minimum work expression may be reduced to a more convenient form for actual calculations. We shall consider that the large system has been divided into two parts: a small part, in which a fluctuation occurs and sta¬ tistical equilibrium is spontaneously disrupted, and the remaining part, in which the variation of quantities is reversible. In other words, the fluctuation has produced a deviation from equilibrium only in a small part of the system. The quantities referring to this part wiU be written without any indices, while those relating to the entire remain¬ ing part will be primed, and equilibrium quantities will be written with 0 index. By definition, the minimum work is calculated in the case of constant entropy of the whole system; i.e., as if instead of a fluctuation occur¬ ring there is some change in the quantities at the expense of external action which does not destroy the statistical equilibrium. Given external action, the work is equal to the change in the energy of the system: Amta=AZ+A^'. (48.7) The work here is equal to the energy change taken with positive sign (48.7) because, by definition, Amin is performed on the system. Sec. 48] FLUOTCATIONS 661 The changes in the quantities in the large system are very small, being less the larger the system, so that A S' may be replaced from the thermodynamic identity (46.26); Ar =0oA/S' —PoAF'. (48.8) As already pointed out, Amin is calculated in the case of reversible process. Therefore, A8'=—AS and, in addition, AF'=—AF, of course. Hence, Ajain = AS — %AS + PoAV. (48.9) Large fluctuations are highly improbable. Therefore, the quantities AS and AF should be regarded as small in the subsystem also; but it is now necessary, here, to make a series expansion up to second order quantities since, otherwise, Amin, would be identically equal to zero (close to the maximum, the entropy expansion can begin only with quadratic terms): + W ( 3 ^ \ IB ^ \ — dg, = —Po> quadratic terms remain in the expression for Aniin=(A^ 4-A if'). These terms may be repre¬ sented in somewhat diflerent form. Taking advantage of the fact that 'ep\ . __/89\ . _ /ap\ —l^^\ Jvfs’’ {8s^ffr~{as)y’ dvas ~ \8sly~{avls’ we write Amin as And so Einstein’s formula for fluctuation probability is transformed thus: where the 0 index is omitted from 9. Let us fin d the probability for volume and temperature fluctuations. To do this we replace Ap and AS by their expressions in terms of volume and temperature; 652 STATISTICAL PHYSICS [Part IV But according to (46.39) right-hand side of equation (48.11) is represented as the product of two factors de¬ pendent only upon AF and A6: (48.12) It is now' easy to determine the mean square fluctuations (A F)® and (A0)*. For the time being we write Then the square of the volume fluctuation is easily written in the form 4-00 (AF)2 = ~ InJe-X^nVlAF): 8a In 1 2a (48.14) The integration was justifiably extended from the integi'and is very small for large A F. We finally arrive at the formula (AF) 2 _ i^P\ \dV)o -oo to oo, because (48.15) It must be remembered that this is not a volume fluctuation, generally, but only at constant temperature. At constant entropy, for example, the expression would have been different. The square of the tempera¬ ture fluctuation, too, is found analogously: Gv • (48.16) This fluctuation is calculated for constant volume. We notice that the square of the fluctuation of volume is directly ldv\ since proportional to the first power of the additive quantity {^~gp volume is an additive quantity. Hence, the relative volume fluctuation ]/(AF)® /F is inversely proportional to the square root of the dimen¬ sions of the system. This statement, as applied to energy, was ex¬ pressed in Sec. 45. The temperature fluctuation y(A0)® is inversely proportional to the square root of the specific heat and, for this reason. Sec. 48] FLUCTUATIONS 553 naturally, also decreases together with the dimensions of the sub¬ system. The quantity 6 is the modulus of the Gibbs distribution for the entire large system. When a fluctuation occurs in the subsystem, 0, naturally, does not coincide with its temperature, i.e., with its distribution modulus, which refers to the time interval during which the sub¬ system is quasi-independent of the largo system. During such a time interval, 0 is not the temperature of the large system either, since 0 has the meaning of temperature only in equilibrium. The temperature and energy of a system are not related quite im- ambiguously: at a given energy, temperature can experience slight fluctuations, and at a given temperature, the energy fluctuates. Thermodynamic inequalities. I^om formulae (48.15) and (48.16) there follow the very important thermodynamic inequalities: (^)^<0,Ok>0. (48.17) The state of a substance can be stable only when these inequalities are satisfied. If the equation of state of a substance indicates that these inequalities break down ftt certain p, F, and 0, then the sub¬ stance is unstable for such p, F, and 0 and must break up into separate phases (liquid and vapour, for example), to which other values of F correspond. The mean product of the fluctuations of two quantities. Let us now consider together the fluctuations of volume and entropy. In this case, the formula for fluctuation probability looks like this: = e ' ‘ ® (Ty-)s. (48.18) Here, the expression on the right-hand side no longer separates in¬ to the product of two factors that depend on each variable separate¬ ly. Therefore, besides the volume fluctuation at constant entropy, and the entropy fluctuation at constant volume, the mean value of the product of their fluctuations also differs from zero: AF A/S^O. Let us calculate this mean from formula (48.18). We write (48.18) in shortened notation: W'-^e ^ . (48.19) In this notation, the required quantity appears thus: AFA6' = - DO DO — OO —-OO In order to calculate the integral, we write the quadratic expression in the exponent in the form of a sum of quadratic terms S64 STATISTICAL PHYSICS [Part IV «u «22 " ^12 ^\2 «Il After this we change the variables in the integral, denoting “11 The integration variable 5 varies within the same limits as A F and A»S', i.e., from —cxj to oo. The integral in (48.19) is / «H«W- an* Oil V „. r a^j r aiia22 otu V “110^22—“12 From this we obtain the required mean value: (48.20) (48.21) We shall now show that this mean quantity is nothing other than —6(-|^)^=0(-|^) . We consider the inverse quantity But if the pressure is represented as a function of entropy and volume in the form p=p [8, V (8, 0)], then the latter expression is — whence it follows that AVA8 The volume and entropy fluctuations are said to be related, or cor¬ related. This is understandable, since if the volume of a system increases, then the statistical weight of its state (i.e., its entropy) also increases. Scattering ot light by fluctuations. Because of fluctuations, no medium can be completely homogeneous. For this reason, electro¬ dynamical equations, for which the constants of the medium s and x Sec. 48] IXTJCTUATIOlfS 656 are regarded as being fixed and everywhere the same, are, strictly speaking, nowhere valid. There always exist small imperfections in the homogeneity which must affect the propagation of light in the medium. Plane waves cannot be propagated in a nonhomogeneous medium; fluctuations cause scattering oif the waves. Let us consider the quantity of scattered energy as a function of the frequency. We shall consider that only the dielectric constant e experiences fluctuations (since % is always close to unity in transparent media). The wavelength of visible light X is about half a micron, which is considerably greater than the mean dimensions of regions in which any noticeable fluctuations occur; this is because very many mole¬ cules are still contained in subsystems of volume cm® ~ X®. and even in gases under normal conditions. The period of oscillation in a light wave is of the order 10“^® sec, and is considerably less than the time during which fluctuation occurs. A time of at least 10-^"= 10-^^ sec is required for the establish¬ ment of statistical equilibrium in the very smallest subsystem. For example, under normal conditions, the time interval between two collisions of a gas molecule is about 10~® sec, and there is absolutely no reason for equilibrium to be established in the condensed phase a million times faster. The velocities of the molecules are very close in aU phases at the same temperature, and interaction, in establish¬ ing equilibrium, must be transmitted over distances not less than 10-« = 10-^ cm. We can, therefore, consider that in the region where fluctuation has occurred the parameters of state of the medium, including polar¬ izability, have changed somewhat. The polarization of the medium, produced in this region by a harmonic light wave, depends upon time in accordance with the same law that determines the electric field of an incident wave, i.e., like Since the dimensions of the region are very much smaller than the wavelength of the in¬ cident light, the polarization has the same phase over the whole region. Consequently, the polarization may be integrated over the whole region, so that a resultant dipole moment is obtained propor¬ tional to The light scattering problem must be considered here to a dipole approximation, in accord with the condition r X being satisfied (see Sec. 19). The total scattered energy is proportional to the square of the second dipole-moment derivative, i.e., a* or provided we neglect the way that polarizability depends upon frequency. When light from the sun passes through the earth’s atmosphere, the blue rays are scattered more than the red, because the wave¬ lengths of the blue rays are shorter. Therefore, the blue portions of the solar spectrum predominate in the scattered light from the sky. This explains the colour of the sky. 656 STATISTICAL PHYSICS [Part IV Exercises 1) Write down Poisson’s formula (48.2) for largo N and N. We represent (48.2) as _ _ __ I_. -N+NInN-NlnN + N where JV! is written to the same acciu'acy as in exercise 1, Sec. 39. N I N — N\ Further, wo must express Inas — ln|l -1- ^—-j and e.xpand in a series up to the scconil term inclusively. Tliis loads to the (.iaussian distribution: _ 1 (n-n)° tt’A" = —e ^ , {AN)^ = N. The same value {AN)^ is obtained from the exact Poisson formula if we write A’2 MW == c' ^ A" -4= A--.^ y 4^ A-Le'^ = . A ^ A! aA aA = A2 -I- A ; A2 - A2 = A. 2) Fmd the pressiHO fluctuation for constant entropy, and (he entropy fluc¬ tuation for constant pressure. Answers: rA6y = CV; (Ap)'^^-- . 3) Find the mean value of AOAp. Answer: AOAp(-If.) 4) Find the fluctuation of the energy and the number of fpianta for an electro- mognotic field of given frequency. Proceeding from the expression h o> It (O e 9-1 we ol>tain, with the aid of the formula derived in Sec. 45 (45.22) h 1 . is the number of molecules in imit volume, p is the polarizability of a single molecule. The additional dipole moment introduced in some volume V by the density fluctuation is equal to d^Ee(N L’o • ^N . 4 nn The square of its time derivative is ~e-l 4 nn After averaging over fluctuations, we obtain dfi = a*ElN z — 1 4 ntt The attenuation in the energy flux of a plane light wave over unit length is equal to 2 /c r.2_ (e-l)2 _ 8,.Me:-l)^ 3 c»k/ 4,t “ n 3 Sec. 49. Phase Equilibrium Separation into phases. A substance consisting of molecules of a single type is characterized by four quantities: the number of par¬ ticles, the temperature, the pressure, and the volume. Only three of the four quantities are independent, since the equation of state must always be satisfied. Thus, for an ideal gas the Clapeyron equation pV=NQ Lids. 668 STATISTIOAI> PHYSICS [Part IV An ideal gas uniformly fills the whole of its permissible volume and in this sense is more an exception than the rule. Thus, for example, if we take one gram of water at a temperature of 20° C, then, no matter what the positive pressure, it is impossible to make it uniformly oceupy a volume 10 cm® (concerning negative pressures, see below in this section). One gram of water at 20° C placed in such a volume separates into two parts—liquid and gaseous; in other words, it does not remain homogeneous. And a certain, very definite, equi¬ librium pressure is established in the system. In the state of statistical equilibrium, the mean number of mole¬ cules going from water to steam in unit time is equal to the mean number of molecules going from steam to water. It will be seen immediately that this condition cannot bo satisfied for all pressures: the number of molecular impacts against the liquid surface is directly proportional to the pressure, whereas the number of evaporating molecules dexiends very weakly upon the pressure. Therefore, at a given temperature, only one pressure corresponds to equilibrium between liquid and vapour. Under other conditions, separation may occur into a liquid and a solid, into a gas and a solid, or into solids of various crystalline modifications or, in general, into phases. The condition for phase eqnilibrinm. Equilibrium pressure can be determined by the methods of statistical physics and does not require a detailed examination of the transition from one phase to another. In the equilibrium state, the temperature and pressure in both phases are, of course, the same. This condition is necessary though not sufficient for equilibrium. In addition, a sufficient condition is that the thermodynamic potential be a minimum (see Sec. 46). The termodynamic potential is additive: it is equal to the sum of the potentials of both phases, and the condition of it being a minimum is written as follows: dO = dOi-h dOg. (49.1) For a given temperature and pressure, the entire change of 2 can occur only due to a change in the number of particles: dd>2 = jXjdAg. (49.2) But as many molecules leave one phase as enter another: dN^ = -dN^. Whence it follows that (Pi — (Xg) dA'i = 0. (49.3) Since dN^ is any number, the phase-equilibrium condition consists in the equality of chemical potentials; t^l (P, 0) = 1^2 (P, 6). (49.4) Sec. 49] PHASE EQUIUBBUTM 559 This equation may be represented in the form of a curve in the p, 0 plane. In other words, to a certain temperature there corresponds very definite pressure. And three phases of the same substance can occur in equilibrium. In this case the equilibrium condition is P-i (P. 6) = H-a (P. 0) = (P. 6)- (49-5) These two equations define a single point in the p, 6 plane (the triple point). Out of it come the equilibrium curves between each two of the three phases (see Fig. 66). Heat of transition. Usually, two phases of the same substance differ greatly from one another; their specific volume, entropy, energy, and other additive quantities experience a discontinuity at the transition point. Let us find the quantity of heat released (or absorbed) at the transition point. Since the transition occurs at constant pressui-e, the quantity of heat is equal to the change in the heat function. We shall refer this heat to a single molecule, so that the heat function must also be referred to a single molecule. Such a heat function will be denoted by i (to distinguish it from I) while the entropy referred to a single molecule will be denoted by — s. The heat of transition to a single molecule is correspondingly equal to q = i2 — ii. (49.6) The heat function is connected with the thermod 3 niamic potential by the relation /—0+6/S. Going over to quantities which relate to a single molecule, and applying (46.48), we obtain i = [i + 0s. (49.7) Whence g'=[X2-fil + e(52-5l). But in equilibrium so that the heat of transition is equal to the temperature multiplied by the entropy change: g' = e(«2 —Si). (49.8) This result is quite understandable since phase transition is a rever¬ sible process. The Clausius-CIapeyron equation. Let us consider two phases of the same substance occurring in mutual equilibrium. We suppose that the temperature in the equilibrium system is changed somewhat. It is required to determine how the pressure must be changed so as to keep the phase equilibrium intact. In other words, the derivative must be determined along the equilibrium curve. 660 STATISTICAL PHySICS [Part IV The dependence of equilibrium pressure on temperature is given in the form of an implicit function (49.4). Hence, the derivative is found according to the usual rule; From (4(5.48) 8 (III — dp , _ 8 <)_ _Jp , (49.9) dp Jo \do)p (49.10) where v is the volume referred to a single molecule. Multiplying the numerator and denominator of the light-hand side of (49.9) by 0, and making use of (49.8), we obtain the required equation: dp ^_ q _ (io 0 — vi) ’ (49.11) which is known as the Clausius-Clapeyron equation. Let us assume that a transition is considered for which q is positive, for example, fusion. Then tlie sign of the derivative is dependent upon which phase has the greatest specific volume; liquid or solid. For example, the specific volume of water at the melting point is less than the specific volume of ice, so that is a negative quantity. If the pressure above an equilibrium system of water and ice is raised, the melting temperature falls. In the transition to the gaseous phase (vapourization, if the transi¬ tion is by a liquid, or sublimation, if the transition is by a solid body) we have the inequality Neglecting in equation (49.11) and reifiacing by --, we obtain dlnp din 0 (49.12) This derivative is always positive. Therefore, the water equilibrium curves close to the triple point may be represented approximately as shown in Fig. 65. The equilibrium curve between water and ice has a negative derivative in accordance with what has been said. Van der Waals’ equation. We shall now show how, from the equation of state of a substance, it is possible to ascertain the necessity of a phase transition. It is convenient for this to make use of the well-known van der Waals equation of state for “real gases.” This equation cannot strictly be derived from the fundamentals of statistical mechan- Fig. 65 Sec. 49] PHASE EQtriLIBBIUM 561 ics on any assumptions, and neither is it supported by accurate quantitative experiments. Nevertheless, it is the simplest of the equations suited for a qualitative description of a very wide range of states, from an ideal gas to its condensation into a hquid. Let us remind ourselves how the van der Waals equation is formed. It is first of aU assumed that a gas cannot be compressed indef¬ initely, hut only to a certain volume 6, which is related to the charac¬ teristic volume of aU the molecules. This is taken into account in the Clapeyron equation pF==N 6 by putting V—b instead of V (in actual fact this has no strict foundation, even if the molecules are regarded as solid spheres). Over large distances between mole¬ cules the acting forces are those of attraction, which fall off rapidly; ui the absence of such forces, condensation into a hquid would be impossible altogether. These forces reduce the pressure. The reduc¬ tion in pressure is inversely proportional to the square of the volume occupied by the gas; this can be shown by the following reasoning. The gas pressure on a wall is proportional to the density of its kinetic energy. The kinetic energy of a molecule incident on a wall decreases due to the attraction of this molecule to the other molecules occmring in the volume. This attraction is due principally to the couplings of the molecules, because the contribution of triple interactions is slight in the case of small gas densities. The quantity of interacting pairs is proportional to the square of the gas density, or inversely proportional to the square of the volume it occupies. Since the energy stems from the attractive forces, its density is negative, and it results in a reduced pressure. The van der Waals equation is iinally written thus: JVO a P— -glTfi — 1^> (49.13) where the second term takes into account the attractive forces. This term in the equation of state can also be rigorously substantiated by the methods of statistical mechanics, but, of course, only at den¬ sities which are still very much less than the density of the liquid phase, where each mol¬ ecule is in constant interaction with many neighbouring molecules. Therefore, the exact equation of state for a real liquid must be immeasurably more complex than the van der Waals equation. It is doubtful whether it is possible to write a single exact equa¬ tion apphcable to a wide class of liquids. Van der Waals’ equation and phase transition. We shall now show how, from the van der Waals equation, the existence of a range of states, in which the substance separates into gaseous and liquid phases, can be demonstrated. Equation (49.13) is third degree in 36-0060 562 STATISTICAL PHYSICS [Part IV volume. It must have three real roots for certain values of 6 and p. In other words, the pressure-versus-volume curve at constant tem¬ perature (isotherm) is of the form ABFD, as showai in Fig. 56. But the derivative is positive between the points B and F and, in accordance with the first inequality of (48.17), the state of the substance is unstable if ^ • Hence, the necessity for the separation of the substance into two phases in this region. The portion of the curve AB corresponds to the liquid state (small volume). As the pressure is reduced the liquid expands to the point A', after which change occurs along the straight line KL. The points K and L are uniquely defined from the equality condition of the chemical potentials (49.4), and the intermediate points along the lino correspond to a mixture of liquid in a state corresponding to J(, and vapour ui a state L. We notice that the position of the pomt A', at a temperature corresponding to the given isotherm, is defined uniquely. The portion KB is not absolutely unstable, since on it < 0 . The states of this portion can be attained without allowing the formation of vapour bubbles in the liquid (a superheated liquid). For this the liquid must be free from foreign agents, for example, bubbles of dissolved gases, which favour vapourization. Sometimes the portion KL lies partly below the abscissa axis, thus corresponduig to a negative pressure, i.e., an extension of the liquid. A liquid can indeed be extended if it adheres eveiywhere to the walls of the vessel and does not have a free surface. The portion FL corresponds to a supercooled vapour, which can be obtained if condensation centres are prevented from forming. Such condensation nuclei or centres easily arise from ions, for example. This is the underlying principle of the Wilson cloud-chamber for the observation of the tracks of charged particles. The critical point. At a sufficiently high temperature, the first term on the right in the van der Waals equation predominates over the second. The equation then becomes very similar to the Clapeyron equation for a volume F— h. But this equation has only one real root for each value of p. This corresponds to the Avell-known fact that at high temperatures a substance does not split into two phases at all. Let us find the temperature at v'hich separation into phases ceases. On the corresponding isotherm A' OD' (Fig. 56) the points B and F, where the derivative j^becomes zero, merge into one point C, and the region of unstable states disappears. All three roots of equa¬ tion (49.13) merge at the point C, so that C corresponds to the triple Sec. 49] PHASE EQUIUBRItTM 563 root of this equation. But the expansion of the function with respect to the difference V—Vc must begin with a third-order term if Fc is a triple root. The linear and quadratic terms in the expansion become zero if the first and second pressure derivatives with respect to volume are equal to zero at the point C. It is easy from this to determine the position of the point G from the van der Waals equation. Let us write down the condition that the first and second deriv¬ atives become zero: / d^p \ From this we obtain NOc _, 2^ „ ' (Fc- ’5)2 F»c ' 2 Oc 6 o „ CVc-bf ' V*c~ Vc-b _ Vc 2 3 ’ Vc = 3b. Then, from (49.14), we find (49.14) (49.15) (49.16) so that _ JV0cF\; ““ 2(Vc-bf = ^lmcb, 00 = 8 a WWb' (49.17) The pressure at the point C is determined from the van der Waals equation: Noc: a \ a yc-b ~ “fF (49.18) If we represent the phase equilibrium curve in the p, 0 plane, then this curve will end in the point p = pcj0 = 0c. G is called the critical point. Separation into phases does not occur at temperatures 0 > 0c. The critical point can exist only on the equilibrium curve between two such phases, which have no feature that is incapable of varying continuously. An example of such a feature is the regularity of crystal structure: in principle, the position of an atom in an ideal crystal defines the position of the whole crystal (with the exception, naturally, of its orientation in space). Yet, the position of an atom in a liquid affects only the position of its closest neighbours. And so for certain substances a continuous transition between the solid crystalline phase and the liquid phase is impossible. The curve dividing the crystalline and liquid phases cannot end and cannot, therefore, have a critical point. 36* 564 STATISTICAL PHYSICS [Part IV The law ol corresponding states. Eliminating the constants o, h and N with the aid of (40.16), (49.17), and (49.18), we have 6 = J^,a = 3Fc®2Jc,iV=| pc Vc 9c (49.19) The last of these three equations shows by how much the equation of state of a substance differs, at the critical ])oint, from that of an ideal gas: pc Vc = -^NQc- But we should note that, as a general rule this relationship is not really satisfied. As has already been mentioned, the van der Waals equation is qualitative in character, and so there is nothing surprising in the fact that for real substances pc Fc iV^6c. If we now substitute (49.19) in (49.13), we get P 80/0c o/^)' PC “ (3F/Fc)-1 \Vcr (49.20) Formula (49.20) expresses a special form of the so-called law of corres¬ ponding states: for two different substances, the ratios and —are related by a single universal equation. It should be noted that in general form the law of corresponding states, especially for substances of similar structure, is satisfied better in practice than the specific formula (49.20) based on the interpolation van der Waals equation, because this, more general, law does not impose a definite functional form on the equation of state. However, there are, of course, deviations from the law of corresponding states also: the ratios are not strictly the same for two substances having identical — and —. pc Oc The properties of a substance close to the critical point. Let us now investigate the properties of a substance close to the critical point in general form, without assuming that the van der Waals equation (49.13) holds. We shall only make use of the fact that the derivative (-“jo, close to the critical point, must tend to zero like the square of the difference F—Fc, because the expansion of p on the critical isotherm begins with a term proportional to (F—Fc)®. At temperatures suffi¬ ciently close to critical, differs from first-order quantities in (6—6c), because the relationship between pressure and temperatm’e close to the critical point does not exhibit any peculiari¬ ties. Thus, the expansion in the critical region is of the form (|f),=-X(F-Fc)®-v(e-9c). (49.21) Sec. 49] PHASE EQUILtBRnm 566 Here, v >0, because the inequality 0 must always be satis¬ fied at temperatures higher than critical. Therefore, X > 0 also. At temperatures lower than critical, becomes zero at two points. They correspond to B and F in Fig. 56 Fb—F c=-l/^(0c-e), Ff-70 = 14^(00-0). (49.22) Tjct us now find the points on the isotherm that correspond to K and L, i.e., to phase equilibrium. For this we make use of the phase equilibrium condition [XK=tXL. It is conveniently written in the form of an integral taken along the isotherm on wliich the points K and L lie: Cdfi. = 0. (49.23) k Multiplying by N and then replacing d by V dp for 0 = const, we obtain L L L jdii=jVdp=j(V-Vc)dp, K K K L because the integral jdpis equal to pt —PK and becomes zero accord- K ing to the condition pL=PK- Now substituting the initial expression (49.21) we see that the equality of chemical potentials reduces to the requirement |(7- Fc) [X(F- Fc)* + V (0 - 0c)] dV= 0. (49.24) The mtegrand is odd in F—Fc. Therefore the integral becomes zero, if at the integration limits (i.e., Fk—F c and Fx,—Fc) the values of F—Fc are equal and opposite in sign or, in other words, the volumes Vl and Fx differ equally from critical. We also represent the condition of pressure equality in integral form X. ^'x. Jdp=||f-dF=0. (49.26) K Vg; Substituting (49.21) here and integrating, we obtain 566 STATISTICAL PHYSICS [Part IV I ( Vl~ Fc)» + V ( Fl- Fc) (6-0c) - y ( Vk - Fc)^ - -v(Fjc-Fc) (0-0c)=O. Making use of the fact that Vk—V c — — (Fl — Fc), we obtain the re¬ quired equation 3 ( Fl - Fc)» + V ( ri. - Fc) (0 - 0c) = 0 , from which it follows that Vl - Fc = Fc - Vk = . (49.26) Thus, close to the critical point, the region of absolutely unstable states is narrower. in the ratio , than the whole region where phase sejiaration occurs. Let us now find the heat of transition close to the critical point. By definition we have Q^Qc (6'l - Sk) = 0c ( Fl - Fk) . (49.27) dS At the critical point, the derivative ypr maintains a finite value l^it is equal to • Therefore, the heat of transition is proportional to (0c—0). Right at the critical point it becomes zero, as expected. Close to the critical point, the density of the substance experiences large statistical fluctuations, because density fluctuations are inverse¬ ly proportional to as we have seen in the previous sec¬ tion, this results in a strong scattering of light. As a result of this scat¬ tering, the substance acquires a certain turbidity, similar to the tur¬ bidity of opal (critical opalescence). Phase transitions of the second kind. At the phase transition point the thermodynamic potentials of both phases are equal. The other additive quantities (such as entropy, energy, and volume) experience discontinuities. But there also exist phase transitions for which not the additive quantities themselves are discontinuous, but only their derivatives—specific heat, compressibility, etc. An example of such a transition was already given in Sec. 43; this is the transition of helium at a temperature of 2.2° K. The specific heat at the transition point changes discontinuously. Another example is the transition of iron from the ferromagnetic to the nonfeiTomagnetic state at 770° C (the Curie point). Phase transitions of the second kind are very frequently observed in crystals. In this case they correspond to a certain change in the translational or vibrational symmetry of the lattice. Since the form Sec. 49] PHASE EQUILIBBHT.M 667 of the symmetry cannot change continuously (the symmetry property either exists or it does not), symmetry always changes disconti- nuously. If an entropy discontinuity is then experienced, we have a phase transition of the first kind; if the entropy is continuous, and the derivatives experience discontinuities, the transition is of the second kind. Let us interrelate the derivative discontinuities of various quantities on the lines of phase transitions of the secopd kind. Since entropy and volume are continuous, we write A,S' = .S 2 - A'l - 0; A F = Fa - = 0. (49.28) Let us differentiate these equations with respect to temperature along the transition line. VVe then obtain Here ^ denotes the derivative of pressure with respect to temperature along the transition curve. Further, = — (v^) I-®®® (46.46)] and Whence, after eliminating A obtain (49.31) Thus, along phase-transition lines of the second kind, the specific heat discontinuity at constant pressure is associated with a compressibility discontinuity. A similar expression can also easily be found for the discontinuity in specific heat at constant volume. Sometimes, phase transition lines of the first kind become a phase transition line of the second kind at some point. If the transition is associated with a change in symmetry, then neither line can simply terminate. The thermodynamic theory of phase transitions of the second kind has been developed by L. D. Landau (see L. D. Landau and E. M. Lif- shits, Statistical Physics, Gostekhizdat, 1951.) Exercises I) Find the specific heat of one of the phases of a substance along the curve of phase transitions. From the definition of specific heat c=e es 00 dp ies\ - do \0p /e. = Cp __ 7__/0T\ 0{F,-Fi) \00/p- 668 STATISTICAL PHYSICS [Part IV 2) Show that Cp becomes infinite at the critical point. Use the result of exercise 4, Sec. 46, and the condition deiining the critical point. 3) Find the discontinuity in specific heat (expressed in terms of compressi¬ bility discontinuity) at constant volume along a phase-transition line of the second. Sec. 50. Weak Solutions Weak solutions and ideal gases. Weak solutions exhibit many regu¬ larities which make them similar to ideal gases. The reason for this similarity can bo easily seen in the fact that the molecules of a dis¬ solved substance in a weak solution interact just as little as ideal gas molecules. But the molecules of a dissolved substance interact strongly with the surrounding molecules of the solvent, whence the differences between a solution and a gas. The thermodynamic potential of a weak solution. We shall proceed from the general expression for free energy in classical statistics: Jf’=-einj'e 8 dr (50.1) [we have omitted the unessential factor (2 tcA)^]. The integral is taken over all physically different states of the system. If one takes into account the identity of aU theiV^ molecules of the solvent and the w mole¬ cules of the dissolved substance, he can extend the statistical integral over the entire phase space and divide it by the total number of per¬ mutations of all identical particles. The number of such permutations is N\ n\. We now write down the thermodynamic potential of a weak solution (I) = jF-f pF=(-61nje“'8dr-f einiV^! -[- pF)-f01nM!, (50.2) where the integi’al extends over the whole phase space of the system. Let us expand the expression in brackets, on the right-hand side of (50.2), in powers of the small quantity , taking into account that the zeroth term in the expansion is the thermodynamic potential of a pure solvent O,,. In addition, we replace In w! by win — according to Stirling’s formula: = + nOln^. (60.3) We can refine the dependence of B (p, 6, N) on the number of solvent particles N by noting that the thermodynamic potential must be an Sec. 60] WEAK SOLUTIONS 569 additive function of N and n. In other words, if N and n increase a certain number of times, for example, twice, must also increase by a factor of two. But in (60.3), this requirement is satisfied directly only by the potential of the pure solvent o, equal to N (Xq, where [Xq is the chemical potential of the pure solvent. For the second and third terms to be additive, let us first write the third term in the form 6wln — = 0wln + 0 m In iV^. e eN After this, the thermodynamic potential will look like «> = [Xo (p, 0) + «01n -^ + n + 01n a) . In order to obtain an additive expression, wo must demand that the function ^ + 0 In iV should not be dependent on N at all. The result is a general expression for the thermodynamic potential of a weak solution: O = iV[Xo(p,0)+n01n-^ + »X(p,0). (60.4) The chemical potential of a solvent in solution is equal to _ 3 ® _ ^ ~ dN ~ 1^0 ■ n9 “aT while the chemical potential of the solute is V- 8 ® 8n 01n-^ + X(p,0). (60.6) (60.6) Osmotic pressure. Certain semipermeable membranes pass solvent molecules freely, but do not pass molecules of the dissolved substance. The solvent must be in statistical equilibrium on both sides of such a membrane. But this is possible only when the chemical potentials of the pure solvent and the solvent in the solution beyond the membrane are equal. The temperature of the substance on both sides of the mem¬ brane is, of course, the same, for otherwise equilibrium could not set in. Only the pressure can differ, provided the pressure difference is held in check by the membrane. Denoting the pressure difference by Ap, we obtain the equilibrium condition: H-o (P. 0 ) = {X (p -f Ap, 0 ) = (Xo (P + Ap, 0) — . (60.7) Let us expand [Xq in a series in powers of Ap, to the linear approxi¬ mation. This expansion is justified, since the pressure difference for a liquid Ap is a small quantity. Therefore, lXo(p + Ap,0) = ixo(p,0) + 4^Ap. (60.8) 570 STATISTICAL PHYSICS [Part IV But the derivative is equal to the volume of a single molecule of pure solvent: ^l-tQ _ _P_ dp N Whence we obtain the equation V • Ap = nO. (50.9) The excess pressure Ap in the solution is called the osmotic pressure. Kijuation (50.9) boars a striking resemblance to the Clapoyron equa¬ tion for ideal gases. It was originally found experimentally and served as the basis for the formulation of a thermodynamic theory of solu¬ tions. We obtained equation (50.0) by proceeding from the general jirinciples of statistics. Phase equilibrium of a solvent (Baonlt’s laws). We shall now consider another case, when equilibrium is also established between solvent molecules. Lot the solution occur in equilibrium with another phase of the solvent, while the solute does not pass into this phase. We find the displacement of the phase equilibrium curve in the p, 6 plane. I^et us call the chemical potential of the phase into which the solute does not pass, g,. Then the phase-equilibrium condition of the pure solvent is determined by the equation (Xi(p,0) = [Xo(P,9), (50.10) while the equilibrium of the other phase of the solvent and solution is ilisplaced and is given by the following condition: gi (p ~h A'p,0 -f AO) = po(P + Ap,0 -p AO) —(50.11) Let us expand the chemical potentials in a series in Ap and AO: [ji, (p 4- Ap,0 -f AO) —[io(p 4- Ap,0 -h AO) = (Xj (p,0) — go (p, 6) + + [-^y (t^i—!^o)] Ap + ((Ai—(Ao) ] AO = = («i —*’o) Ap —(«! —«o) AO. (50.12) We shall now assume that the pressure in the system is the same as above a pure solvent, i.e., that Ap = 0. Then the equilibrium-temper¬ ature displacement AO will be defined: "TT’ (50.13) where Q—NO (s^— Sq) is the heat of the phase transition of the pure solvent. For vapourization Q>0; therefore, AO>0 if the solute does not pass into vapour, so that the equilibrium temperature is raised. Indeed, the solution has a higher boiling point than the pure solvent. Sec. 50] WEAK SOLUTIONS 571 Let us now suppose that the solute does not pass into the solid phase of the solvent. Then Q is the heat of solidification, Q < 0. It is seen from this that the fusion temperature of the solution is lower than that of the pure solvent. The use of cooling mixtures is based on this prop¬ erty of solutions. Let us now consider equilibrium at a given temperature, A9 = 0. Then the reduction in equilibrium pressure over the solution is deter¬ mined from (50.12): If a solution is in equilibrium with vapour, then Vq. The product Ni'i is the volume of the entire solvent in the vapour state. If it were possible to transform the solute to vapour together with the solvent, the partial pressure of the molecules of the substance would equal the reduction in the equilibrium pressure above the solution. The relative pressure reduction —A p/p is equal to the concentration of the solu¬ tion nIN. Solute equilibrium. A solution is termed saturated if it is in equilib¬ rium with the dissolved substance. The equilibrium condition con¬ sists in that the chemical potential of a pure solute, [i'q, is equal to its chemical potential in the dissolved state: ^'„ = ji' = ein-^ + X(p,e). (50.15) We have supposed that the saturated solution is also stiH regarded as weak, i.e., If a pm^ substance occurs in a gaseous state, then its chemical poten¬ tial depends upon pressure according to the law [see (47.17)] (i'o=01np-[-/i(6). (50.16) The function x (p, 6) is but slightly dependent on the external pressure: X (p, 6) is determined by the properties of the condensed phase, which do not change when the external pressure varies over several atmos¬ pheres. Comparing (50.15) and (60.16) and taking antilogarithms, we find that the equilibrium concentration of the dissolved gas is propor¬ tional to its pressure above the liquid (Henry’s law) ^ = a(0)p. (60.17) The coefficient of p depends very weakly on pressure. Heat of solution. The heat of solution is equal to the difference in the heat functions of the substances comprising a solution before and after being dissolved. The heat fimction is related to the thermo¬ dynamic potential in the following way: 672 STATISTICAL PHYSICS [Part IV / = O —6 — 62 as Therefore, the heat of solution is equal to (50.18) g = _e2-^-L(iV(io + TC61n^+ nX — nii'o— N , (50.19) where (i'q is the chemical potential of the dissolved substance. The quantities appearing here may be expressed in terms of the concentra¬ tion of the saturated solution nJN with the aid of the saturation condition (50.15). This yields Q = _«G2~ln-^. (50.20) The heat of solution for one molecule is equal to Qjn—q, or 8 0^1 ^ In-— * do n. Wq do (50.21) Thus, if the concentration of a saturated solution increases with tem¬ perature, then heat is absorbed in dissolution. The Le Chatolier-Brann principle. Let us suppose that heat is supplied Q yh to a saturated solution in equilibrium with the solute. Then, if > 0, part of the substance will further dissolve, and the heat is spent not only in raising the temperature but also in dissolving. But if - >. 7 !® < 0 , then some of the substance comes out of solution, on which, do in accordance with (50.21), heat is also expended. In both cases, changes occur in the equilibrium system that counteract the external action (raising of the temperature). The foregoing example illustrates a general rule, known in thermodynamics as the Le Chatelier-Braun principle. The Clausius-Clapeyron equation can be examined on the basis of this principle. The phase rule. Let us suppose that there are k substances (compo¬ nents) distributed in the form of solutions of arbitrary concentration over / phases. How many parameters define the equilibrium state of such a system ? The chemical potentials of the substances depend upon temperature, pressure, and relative concentrations. The concentrations of aU the substances in any phase satisfy the equations 1-1 since, by definition, the concentrations are equal to (50.22) Sec. 60] WEAK SOLUTIONS 673 cf f m The equilibrium conditibn consists in the equality of the chemical potentials of each of the k substances over all / phases: [i} (p, 8, cj cj, ..., Cfc) = (JL? (p, 6, cj, Ca,..., cl) = ... pi (p, 0, c[, ci, cl) ftl (p, 6, cl, cj,. -., cj) =... = pi (p, 0, c(, clcl). (60.23) Here the superscript always denotes phase, while the subscript de¬ notes the substance. Equation (50.23) involves k concentrations in / phases, and two other variables (temperature and pressure), so that there are kf-i-2 variables in all. There are /— 1 equations (50.23) for each substance and, in addition, the concentrations satisfy / equations (50.22), so that in all there are k (/—1)-|-/ equations for determining k/-h2 variables. The number of independent variables which may vary arbitrarily is equal to the difference between the number of variables and the number of equa¬ tions, i.e., r^kf+2 — k(f~\) -j=^k — i + 2. (50.24) The quantity r is called the number of thermodynamic degrees of freedom of the system. (50.24) expresses the Gibbs phase rule', the number of degrees of freedom is equal to the number of components, minus the number of phases, plus two. For example, if a single substance occurs in equOibrium in two phases, then r = 1; in such a system, one may change arbitrarily a single variable: temperature or pressure. In a two-component, two-phase system, there are two degrees of freedom: the component concentra¬ tion in one of the phases can be varied together with the temperature or pressure. Strong electrolytes. The thermodynamic properties of solutions of strong electrolytes exhibit noticeable deviations from the laws obtained in this section for the solutions of neutral substances. It is natural to look for the cause of these deviations in the fact that ions interact electrostatically. This type of interaction was not taken into account at all in the theory of weak solutions. Aqueous electrolyte solutions exhibit very strong dissociation, the cause of which can be qualitatively imderstood if we take into account that the static dielectric constant of water is equal to 81. The potential energy for the atomic interactions of a heteropolar molecule in a solu¬ tion is less than in vacuum by roughly s times, so that in water the atoms of such a molecule are more weakly bound by a factor of 81. 574 STATISTICAL PHYSICS [Part IV Thermal motion in the solution disrupts the bonds, and instead of molecules we have ions in the solution. The interaction forces between ions are Coulomb forces; they faU off with distance much more slowly than the interaction forces between neutral molecules. Let us deter¬ mine the correction to the chemical potential for a weak solution, which correction is due to the interaction forces between ions. The ionic atmosphere. For simplicity, we shall consider an electrolyte which contains only singly charged ions of both signs, for example, H-* and Cl“ in a weak aqueous solution. Both positive and negative ions can occur close to a positive ion. The density of positive ions near an ion of the same sign is reduced, compared with the mean density, by the Boltzmann factor e ® , where 9 is the potential produced by the charge distribution at a given point in the vicinity of the positive ion. The negative-charge density at the same point is increased by the Boltzmann factor v ® . If po is the mean ion density, then the charge density at the point of the solution considered is C?p <'

;-] (50.37) (the factor 2 takes the complete dissociation into accomit). The correction factor inside the brackets tends to unity as the solution concentration rijN decreases, but its derivative with respect to concentration tends to infinity. The additional term involved in equation (50.15), which determines the concentration of a saturated solution, is proportional to Vn. It also contributes infinite additions to the derivatives as n tends to zero. Sec. 51. Chemical Equilibria Reversible and irreversible reactions. Like all processes whose velocities do not coincide -with the rate of change of the external parameters of a statistical system, chemical reactions, which proceed Sec. 51] CHEMICAI. EQUILIBarA .577 with finite velocity, are irreversible. For example, when an explosive luixtnro (hydrogen and oxygen) burns, water vapour is produced irreversibly. If a certain quantity of the oxyhydrogen mixture is prepared in a closed vessel, the state of the mixture will be thermodynamically unstable with respect to the reaction. True, the reaction by no means ]>roceeds directly according to the “gross equation” 2 H 2 + 02 = =-- 2 H^O. To do so, the molecules would have to overcome very high j)otential barriers. In actual fact, the reaction must proceed through stages involving the intermediate unstable substances OH, H, O Avith nonsaturatcd valences; these are the so-called active centres. The initial formation of active centres is very difficult, so at room temperature an oxyhydrogen mixture may be preserved indefinitely. Hut if active centres are somehow produced (by a powerful electric spark, for example) then they are renewed and multiplied in the course of the reaction (a chain reaction).* When the multiplication of active centres is fast enough, the reaction proceeds explosively. But chemical reactions never proceed to the end. If an explosion is ])roduced in a sufficiently strong vessel (bomb), then the finite equilibrium state will contam hydrogen, oxygen, and water vapour in certain (strictly definite) concentrations that depend upon the temperature and pressure and the initial compo.sition of the mixture. This finite equilibrium state is termed ehemical equilibrium. When the state in an equihbrium system is changed slowly, the equilibrium Avill shift in one direction or other, i.e., the quantity of initial or final products may increase. But these chemical reactions ])roceed with the same velocity as that with which the external conditions change. Hence, such reactions are reversible—-as are aU ]}rocesses whose velocity is not estabhshed spontaneously but is all the time equal to the velocity of change of the quantities that the equilibrium state of the system is given by. Chemical equilibrium. The state of chemical equilibrium can be found with the aid of the thermodynamic functions of the substances involved in the “gross equation” reaction, quite independently of the mechanism by w'hich the reaction proceeds. Tliis is Avhy the theory of chemical equilibria had already been formulated in the nineteenth century, while the study of the velocity of chemical reactions is still developing vigorously at the present time. In this sense, the situation is similar to that in statistics in general, i.e., in the science of equilibria, and in kinetics—the science of the ve¬ locities of macroscopic processes. * The majority of chain reactions aro assooiatod with an active centre. This was established by N. N. Semyonov, the discoverer of chain reactions, and his pupils (and independently by C. N. Hinsholwood). 37 - 0060 678 STATISTICAL PHYSIOS [Part IV For a given temperature and pressure, chemical equilibrium is attained only when the thermodynamic potential in the reacting mixture has a minimum dcD = 0. (51.1) When p = const and 6 = const, the minimum condition, d=^\LidNi. (51.2) ( Here, (i,- is the chemical potential of the ith substance appearing in the “gross equation” reaction. For an oxyhydrogen mixture, for example, the only substances of this kind are hydrogen, oxygen, and water vajiour. But the numbers dNi are not arbitrary: they change as the reaction proceeds, and are therefore interrelated by the reaction equation. In other words, Ni may vary only in equiv¬ alent (stoichiometric) quantities. For example, if we take the reaction 2 C 0 -f- O2 = 2CO2, then d Nqo • d No, ’■ d Nco, = — 2 ; — 1 : 2 . In the reaction of the thermal dissociation of hydrogen H2 = 2H d Nh, : dNn — — 1:2. In general, the number d Ni is propor¬ tional to the equivalent of the given substance in the reaction v,; equation (61.2) can also be rewritten thus; 27(x.v, = 0. (51.3) This equation expresses the condition of chemical equilibrium in a system. Law of mass action. Equation (51.3) is especially useful when we have an explicit expression for the chemical potential of the reacting substances, as in a weak solution or in an ideal gas, for example. In the latter case, the equilibrium concentrations of the substances may be determined if there are sufficient data about the structure of all the molecules in equilibrium. The chemical potential for a gas in a mixture of ideal gases is [see (47.17)] [x,- = — 6 In 0 /.-( 0 ) (51.4) where /,• is a statistical sum taken over all momentum values of the moleciile as a whole, and also over aU its rotational, vibrational, and electronic states. The latfer are essential only when they occur close to the ground state of the molecule and are far from its disso- Sec. 51] CHEMICAL EQtrtLIBBIA 679 ciation limit. If they occur close to the dissociation limit, the mole¬ cule decomposes before such highly excited states can in any way affect the values of the statistical sums (see exercise 2 ). Substituting the expression for chemical potential into the chemical equilibrium condition (31.3), and eliminating 0, we obtain ^ Vi In Pi = V/ In 6 fi. I Taking antilogarithms of this equation, we obtain the equilibrium condition (expressed in terms of partial pressures) from the formula □ = (51.6) This equation can also be written in terms of the relative concen¬ trations of the substances by replacing the partial pressures with the aid of (47.15): -S'’.- -S'’i Ci''i = p ' n[( 6 /i)''' = ?> ' K. (51.6) i i Here, ci denotes the coneentration of the ith component of the mix¬ ture: Ci = Ml N • (61.7) The pressure on the right-hand side of (51.6) has still to be expressed in terms of the initial pressure or in terms of the initial density; this can always be done easily with the Clapeyron equation, if we take into account the change in the number of particles (relative to the initial number of particles), for a given equilibrium of the chemical reaction. The component concentrations depend upon the initial quantities of the original substances involved in the reaction. Thus, also the eiiuilibrium concentrations depend upon these quantities (or masses). Therefore, equation (51.6) expresses the so-called law of mass action. The quantity appearing on the right-hand side of equation (61.5) is called the equilibrium constant of the given reaction, because it does not involve the concentrations of the mixture. Its dimension- r ^'■■1 ality is [p' J • Heat of reaction. The heat of a chemical reaction occurring at constant pressure is defined as the difference in the heat functions of the reacting substances before and after the reaction. It is conven¬ ient to write this heat as calculated for a single elementary act of the reaction a 8

0, then, as the temperature increases, the equilibrium tends towards a predominance of those substances that enter into the reaction equation with positive coefficients v,-. The concentrations of these substances appear in the numerator on the left-hand side of equation (61.6). But then, according to (51.10), the system absorbs heat, so that reactions occiw in it which oppose the rise in temperature. The increase or decrease in the temperature of an equilibrium system can produce reversible reactions in it in any desired direction. Exeiciseg 1) Write down the equations of the law of mass action for the reaction 2CO -t- Oj = 2 CO 2 , if a moles of CO and b moles of Oj initially take part in the reaction. Let X moles of Oj react; then 2x moles of CO enter into the reaction with them, and 2x moles of COj are formed. In all, there are a + b — -Zx + 'lx = a + b — x moles of different substances in the system. The concentrations are, re¬ spectively, a — 2x Cco = —Ta— r . a + b — X _ h ~ X “ o -I- 6 — 0! ’ 2x so that the equilibrium equation appears thus: (2 x)‘ (a + b-X) (a-2x)‘(b-x) Here, p is the equilibrium pressure, which differs from the pressure of the original substances p„ (for the same temperature) by the factor ^ Whence a -f- 6 the equation for the required quantity x is PqK (a — 2x)^(b — x) i{a + b) Sec. 61] CHEMICAIi EQUIUBBIA 581 2) Calculate the equilibrium constant for the thermal dissociation of nitrogen, using the following data. The groimd state for a nitrogen atom is ^S. The first excited state lies 2.4 ev higher (*£)), the next lies 3.5 ev above the groimd state (^P). The formation energy of an N, molecule at absolute zero is equal to 9.70 ev (this value is now reliably established). The moment of inertia for the ground state is J = 13.84 x 10~*® gm/cm®. The vibrational quantum of a molecule is equal to 0.287 ev. In the groimd state of the molecule, the orbital and .spin angular momenta of the electrons do not have projections on the line joining the nuclei. The first excited state lies higher than 0 ev above the ground state of the molecule. The statistical sum for the atoms is /n ’ (2 7t WiN 0)it/2 44 - 2 . 5.6 2.4 0 + 2 . 3.6 3^5 'o I Here and in future, it is convenient to express 0 in electron-volts, taking into account that 1 ev corresponds to 11,600°. We shall confine oiu’selves to tempera¬ tures for which the statistical sura for a molecide involves only the electronic groimd state. Then we obtain [see (47.21)]: _ , (27cWfi*0)3/2 2 kJQ 4-n 6 ® /n* = —(2 „A)aT" ■ 1 „ 0 The equilibrium constant for the reaction N 2 = 2N is, from (51.5), equal to K: 111 /n, I 2.4 3.5\a „ , „ / hw\ 9.76 = ( 44 - 106 84-66 9 ) Oe ® ■ To illustrate, let us find the fraction of dissociated molecules, if the tempera¬ ture is equal to 1 ev and there are 2.7 X10^“ molecules in 1 cm®. Then the equa¬ tion of the law of mass action equation is 4 a:® 1 — a: =0.494 . 10« . 5.90 . 10-» = 24.2. Here, the factor before the exponential is equal to ~ 5 x 10®, while the exponen¬ tial is equal to 5.9 x 10~®. The eiiuilibrium degree of dissociation is a: = 0.88. Thus, when the tomperatiu-o is equal to only one tenth of the dissociation energy, 88% of all the molecules have already dissociated. The predominance of the pre-exponential factor over the exponential is explained, for such relatively low temperatures, by the fact that the statistical weight of the dissociated state is determined by the entire volume occupied by the gas, while the nondissociated state is determined only by the volume of the molecules; therefore, at atmospher¬ ic density of the gas (2.7 x 10*® mole/cm®), dissociation is already highly probable. 3) Find the degree of thermal ionization for helium as a function of its tem¬ perature and pressure. Do not take second ionization into account. The first ionization potential of helium is 24.47 ev, while the first excited state lies 20.5 ev above the ground state. The ionization equilibrium satisfies the law of mass action Ce CHe'*' CHe = p-* K. 682 STATISTICAL PHYSICS [Part IV The statistical sums here are: fe- Ieb^ '■ /ho = {2-r:meQpi^ {2nhf ’ (2teWi£o6)®^^ ( 2 rtA)=> 24.47 (2ji miio 0)3/2 _—L (2T:hf (the factor 2 takes into account the spin of the electron and Ho+ ion). From this wo express the equilibritim constant as , (2rtm, 0)3/2 (2itA)> 0 ■ e M.47 6 1 1/2 m’O* F e 24.47 If the initial pressure of helium is Po» then the ionization-otiuilibrium equation assumes the following form: x’‘ _ 1-x ~ po ■ To illustrate, we shall take a temperature of 4 ev and a molecular density 2.7 X 10** mole/cm*. Then the equation of ionization equilibrium is written numerically thus: -T= -8.47 • 103.2.19.10-3 = 7.6. = 0.90. I — X Here, >is in the previous example, the pre-exponential factor predominates over the exponential, equal to ^19 x lO”*, as a result of the largo statistical weight of the ionized state. Excited states of the helium atom make a negligible contribution to the statistical sum. At higher temperatures, the first ionization is practically complete, so that there are simply no neutral atoms which could be excited. 4) Relate the e.m.f. of a primary cell to the heat of the chemical reaction occurring in it. By definition, the e.m.f. is the work done in carrying unit charge aroimd a conducting circuit. If a primary cell i.s connected in the circuit, then, in travers¬ ing it, reversible chemical reactions occur that neutralize the ions at the elec¬ trodes. The work done in a reversible reaction is equal to the change in thermo¬ dynamic potential, while the heat is equal to the change in the heat fimction. Whence, from equation (51.9), we obtain Sec. 52. Surface Phenomena The thermodynamic potential of a surface layer. We have so far considered only the volume properties of a substance, so that all the results relating to phase and chemical equDibria, and to equilibria in solutions, refer, strictly speaking, to very large systems. The surface layers separating different substances or different phases of the same substance exhibit special properties, which, however, depend both on the nature and the states of the volumes in contact. Sec. 52] STTRI’AOE PHENOMENA 583 The thermodynamic potential for unit surface of contact of two media depends upon the temperature 0 and pressure p in the sur¬ rounding media. In equilibrium, 6 and p are constant over the whole surface. Interaction between the contacting portions occurs across their boundaries. The dimensions of the boundaries are proportional to the first power of the linear dimensions of these portions, while their areas are proportional to the squares of the linear dimensions. Therefore, for sufficiently large dimensions, the portions of the sur¬ face layer may be regarded as quasi-independent subsystems, like the way in which we considered volume subsystems. Hence, the thermodynamic potential of a surface is additive for the same reason as the volume potential is. If the thermodynamic potential for unit interface surface of two media is denoted by a and the magnitude of the whole contact surface is X,, then, by virtue of additivity, the potential for the whole surface will be equal to a) = a!:.' (62.1) Surface tension. The work done at constant pressure and temper¬ ature is equal to the change in thermodjmamic potential (Sec. 46). Therefore, in changing surface area by unity, work is performed equal to a. This work is called the surface tension of two given media. It is easy to demonstrate the relation between this definition of surface tension and its elementary definition. Let a film of liquid be stretched on a rigid H-shaped wire frame, closed by a movable member, completing a rectangular liquid surface. If the length of the movable member is equal to unity, then a force acts on it (from the direction of the film) equal to twice the surface tension of the film (because the film has two sides). When the cross-member is displaced by unit length, the force of surface tension performs work numerically equal to double its magnitude. But the total surface of the film increases in this case by two units, so that the work done in increasing the surface by unity is indeed equal to the “force” of surface tension. When the surface area is increased, part of the atoms leave the volume for the surface layer; to do so it is necessary in part to overcome the attractive forces duo to other atoms. This is what explains the origin of the work that is lost (or gained, depending upon the nature of the volumes in contact) in increasing the area. The surface tension of a condensed phase at the boundary with a vacuum is, of course, always positive. The thermodynamic potential is at a minimum when in equilibrium. In this case, the minimum is attained simply for the least area Therefore, a liquid film, stretched on a certain (in the general case a nonplane) frame assumes the least possible surface area for the given frame. A liquid drop in ideal equilibrium assumes a spherical form that has the least surface area for a given volume. 584 STATISTICAL PHYSICS [Part IV The heat ot surface increase. When surface area increases, heat is evolved in addition to the performance of work. Since the process of surface increase is reversible, the heat is determined from the general formula (46.18) Q~0AS. The entropy of the surface is calculated from formula (46.46). Substituting the thermodynamic potential for the surface (62.1) into this formula, we find an expres.sion for the heat; Q = (52.2) doL Thus, heat may be evolved or absorbed depending upon the sign of g—. The equilibrium vapour pressure above a drop. The phase equilibrium condition changes if the surface thermodynamic potential is taken into account in addition to that for volume. Of course, the general condition dj _ 0*2 'aN ~ 'on * (52.3) Let the subscript 1 refer to the vapour phase contained in a large volume, and the subscript 2 relate to a small liquid drop of radius It. Then, for the first phase. aN ~~ (52.4) while for the second phase. 0^2 _ 8N ~ aN (iV (X2 + a Q. (52.5) The derivative of the second term is calculated in the following way; 0 y ax. ^ j^aR (52.6) If the density of the liquid is p mole/cm®, then R = —V’, so that It "pj aR _ I R aN ~ z N ’ (52.7) Substituting this in (52.6) and now expressing N in terms of R, we obtain aN a R^ (52.8) Thus, the equilibrium condition between the vapour and liquid drop is (Xi(p, 6) = {r2(?J.9) + (52.9) Sec. 62] SURFACE PHENOIirENA 585 We represent the pressure f as where Po is i-h® equilibrium pressure on a plane surface. The expansion of chemical potential in powers of Ap gives [cf. (50.12)] (Wi —r2)Ap = -^. (52.10) Neglecting the specific volume of the liquid compared with the specific volume of the vapour, we find a final expression for the excess pressure; The analogous expression for the pressure in a vapour bubble within a liquid leads to the same formula, but with opposite sign. The stability ol supersaturated phases. We have thus seen that the equilibrium vapour pressure over the convex surface of a drop is greater than that over a plane surface, while that over the concave surface of a bubble is less. This explains the relative stability of super¬ saturated phases, which was mentioned in Sec. 49. If a liquid drop appears in a supersaturated vapour with pressui’e p'>Po. whose radius is less than _ p 0 (p' - Po) ’ (52.12) then this drop evaporates once again, and further condensation on it is highly improbable, since it is a fluctuation phenomenon. Only if inequality (52.12) is reversed can the drop begin to grow. But the spontaneous formation of a large drop, like any large fluctuation, is highly improbable. Therefore, condensation usually begins on small nuclei that are already in the vapour, for example, ions. In exactly the same way we can explain why a highly purified superheated liquid does not boil. Boiling of a liquid consists in the formation of vapour bubbles in its volume. In order that such a bubble should not collapse due to external pressure, the equilibrium pressure of the vapour must be at least that of the external atmospheric pressure above the liquid. But if the equilibrium vapour pressure above a plane surface is only equal to the external pressure, then there is not sufficient pressure in the bubble for equilibrium. Therefore, a bubble of insufficient size cannot grow. APPENDIX APPENDIX 687 The summation with the upper sign may be reduced to the summation with the lower sign. Indeed, - 1 -t- + _L. + .. 2(1+1 “ 3"+! 4”+l ~ 1.11 I 311+1 + 411+1 + • • • 41 H •2 • 211+1 I 1 + ~ + 311+1 + • • • j /, 1 \ /, 1 1 , 1 \ (i+2S+r + 311+1 + 411+1 + ... j Finally, the summation involving positive signs has the following values: 1 n^-j 1 3 2 2 5 2 00 =2-612 fe-1 1.646 1.341 1.202 1 1.127 1.0823 For odd n, the following formulae obtain: 00 00 1 _ n* 1 Therefore, fe 1 Wo also note that ■ x^dx e*—1 fe~l 00 00 j-xV^ ^ y . 2.612 = 2.31 . J e* - 1 2 " 2 0 < 1-1 W'e have met with the integral (44.39): 00 00 fe+ ' if(f+e - ^ )- = - 2/** i (i^r) = 0 00 f - 4(1 Je* + 1 i 0 2/ 6 3 — 00 588 BIBLIOaBAPHY BIBLI06BAPHY Part I 1. *L. T). Landau and E. M. Lifshits, Mechanics, Gostekhizdat, Moscow- Loningrad (1958). 2. *0. K. Suslov, Theoretical Mechanics, Gostekhizdat, Moscow-Leningrad (1944). 2 . *T. Levi-Civita and I. Amaldi, Lezioni di meccanica razionale, Bologna (1930). 4. E. T. Whittaker, Analytical Dynamics (1937). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge (1927). 5. A. Sommerfeld, Mechanik (1944). (5. G. Goldstein, Classical Mechanics, Cambridge (1950). Part II 7. *N. E. Kochin, Vector Ccdcidus, GITTL (1934). 8 . L. D. Landau and E. M. Lifshits, Field Theory, Gostekhizdat, Moscow- Leningrad (1948). 9. *1. E. Tamm, Fumlamentals of the Theory of Electricity, Gostekhizdat, Moscow-Leningrad (1954). 10. *Abraham and Becker, Theorie der Elektrizit&t, Bd. I, Leipzig (1932). 11. *P. Becker, Elektronentlieorie. 12. A. Einstein, The Meaning of Relativity, Princeton (1953). 13. P. G. Bergman, Introduction to the Theory of Relativity (1942). 14. *Einstein and Modem Physics, Gostekhizdat (1966). Part III 1.5. L. D. Landau and E. M. Lifshits, Quantum Mechanics, Part 1 (1948). 16. ^D. I. Blokhintsev, Fundamentals of Quantum Mechanics (1944), 2nd ed. (1949). 17. A. Fok, The Principles of Quantum Mechanics (1932). 18. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two- Electron Systems, Springer-Verlag (1957). 19. P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford (1957). 20. E. V. Shpolskii, Atomic Physics, Parts 1 and 2 (1960). Part IV 21. L. D. Landau and E. M. Lifshits, Statistical Physics (1960). 22. *V. G. Levich, Introduction ta Statistical Physics (1954). 23. ♦M. A. Leontovich, Statistical Physics (1944). 24. *M. A. Leontovich, Introduction to Thermodynamics (1950). 25. *E. Fermi, Molecules and Crystals (1947); (Molekiile und Kristalle, 1938). 26. ♦A. G. Samoilovich, Statistical Physics (1955). * As far as the author knows, these have not been translated into English. SUBJECT INDEX 589 SUBJECT INDEX Abberation of light: 200 Absolute thermodynamic scale of temperature: 622 - - electrostatic system of units: 106 — black body: 467 Absorption of light: 170 Action: 82 — for a particle in an electromagnetic field: 218 - for an electromagnetic field: 117 — in the theory of relativity: 211 Additive quantity: 16 — integral of motion: 36 Adiabatic process: 620 ~ demagnetization: 643 Alpha disintegration: 283 Amplitude: 61 Analogy, Mandelshtam: 282 —, optical-mechanical: 235 Angle, Eulerian: 79 — -, solid: 54 Angular momentum: 36 of a field: 124 —, as a generalized momentum: 42 — composition of: 309, 337 Antiparticle: 401 Antiproton: 401 Aphelion: 45 Atmosphere, planetary: 441 Atom, vector model of: 369 Atomic imits: 316 Atoms, hydrogen-like: 322 —, metastable 367 Barrier factor: 282 - for alpha disintegration: 284 Biot-Savart law: 137 Bohr theory of atomic structme: 229 - magneton: 330 ■ quantum conditions: 290 Boltzmann distribution: 430 Born approximation: 385 Bose condensation: 474 Centre of mass: 26 Proper time: 205 Chemical eq^ibrium: 576 - potential: 668 Circulation of a vector: 96 Close action: 134 Coefficient, mutual induction: 160 —, self-induction: 160 Collisions, elastic: 61 —, inelastic: 60 Commutative relations for operator: 298 Condition, Lorentz: 116 Conductors: 149 Constant, equilibrium: 579 Constant, Planck’s: 230 Constraints, ideal rigid: 16 Contraction of the length .scale: 198 —, time interval: 197 Coordinate system: 11 —, centre of mass: 49 —, inertial: 15 —, laboratory: 49 , rotating: 69 Coordinates, curvilinear: 101 • -, generalized: 12 —, normal: 66 , normal electromagnetic field: 275 —, spherical: 24 Coupling, j-j: 339 — ■, Bussel-Saunders: 338 Critical point: 662 Curl: 96 Current, displacement: 111 Cyclic variabfes: 43 Deflection of light rays in a gra¬ vitational field: 210 Degree of freedom: 11 —, thermodynamic: 573 Density, charge: 109 -, current: 110 —, energy: 122 —, probability: 244 —, probability flux: 250 Diamagnetic substance: 161 Diamagnetism of electrons: 485 Dielectric: 149 — constant: 163, 162, 442 Diffraction, electron: 239 —, x-ray: 239 Dipole approximation: 185, 362 Dispersion: 379 — formula: 382 Distribution, Boltzmann: 430 —, Bose-Einstein: 426, 474 —, Fermi-Dirac; 426 —, Gibbs: 602 —, Maxwell: 432, 477 Divergence, vector: 94 Domains: 162 Effect, anomalous Zeeman: 374 —, Compton: 231 —, Dopier: 207 —, linear Stark; 377 —, normal Zeeman: 374 —, square-law Stark: 377 Effective cross-section, differential: 64, 385 —, total: 391 Efficiency: 622 Eigenfimctions: 253 690 SUBJECT INDEX Kigonvaluos: 253 —, angular-momontuinproiection:294 ; degenerate: 416 - of square of angular momentum: 308 Einstein formula for fluctuation pro¬ bability: 550 radiation laws: 464 Eloments, rare-earth: 342 - transuranium: 243 Emission of quanta, spontaneous and forced: 462 magnetic dipolo: 188, 366 , quadrupolo: 188, 367 Energy: 31 -, field: 121 -, hydrogen atom: 410 -, in the theory of relativity: 213 -, of potential gauge calibration: 44 -, potential: 19 -, rest: 213 -, solid body: 467 -, calculation of: 468 - -, total: 32 Energy levels, electronic: 448 - —, fine structm'e of: 330 multiplet: 339 -, rotational: 452 -, spectroscopic notation of: 339 - -, vibrational: 449 Entropy: 506 - — of a subsystem: 507 Equation, Clapeyron: 436 -, Clausius-Clapeyron: 559 Dirac: 395 - -, Schrodinger: 246 —, Thomas-Fermi: 487 - -, van dor Waals: 560 Equations, Euler: 78 Hamilton: 89 —-, Lagrange; 19 —, Maxwell: 109, 113 -, in complex form: 157 Excited state of a system: 254 Expansion, eigenfunction: 304 Experiment, Fizeau: 200 —, Michelson’s: 191 —•, Stern-Gierlach: 295 Factor, Land6: 372 Fermi-Dirac distribution: 426, 477 Ferromagnetism: 151 Filled levels, shell: 370 Pine structure of atomic levels: 330 Fizeau experiment: 201 Fluctuation, absolute energy: 504 —, relative energy: 504 Force: 14 —, central: 24 Force centrifugal; 73 —, Coriolis: 72 electromotive: 106 —, friction: 16 —, inertia: 71 —-, Lorentz: 220 -, magnetomotive: 112 - oscillator: 384 Formula, barometric: 441 -, Debye: 470 --, dispersion: 382 Pl^ck: 464 - -, Poisson: 548 —, Rutherford: 55, 387 —, Stirling: 423 Formulae, Fresnel: 172 Free energy: 624 Frequency, of oscillation: 58 Function, Hamilton: 88 - Lagrange: 22 —, for field: 117 Galilean transformation: 68 Gauge calibration, potential: 115 —■ invariance: 116 Gibbs distribution: 502 Gradient: 98 Ground state, electromagr etic field: 277 - of a system: 264 Hamilton’s equation: 89 — function (Hamiltonian): 88 Harmonic oscillator: 61, 266 Heat of reaction: 679 — of solution: 571 Helium, orthostate of: 349 — parastate of: 349 Ideal gases: 415 Index, adiabatic: 542 —, refractive: 170 Induction, electric: 148 —, magnetic: 148 Integrcd principles: 82 Integrals of motion: 31 Intensity, dipole radiation: 187 Interval: 202 Intervals, space and time: 203 Invariant electromagnetic field: 223 Inversion: 320, 403 Irreversible process: 620 Isomers, nuclear: 367 Isothermal process: 619 Kepler problem: 47 Lagrange’s equation: 22 —, inte^als of motion: 31 Lagrangian: 19 —, field: 117 Land6 factor: 372 Laplacian operator: 100