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MATHEMATICS OF CLASSICAL AND QUANTUM PHYSICS FREDERICK W. BYRON, JR. AND ROBERT W. FULLER Two VOLUMES BOUND AS ONE Dover Publications, Inc., New York Copyright & 1969, 1970 by Frederick W. Byron, Jr., and Robert W. Fuller, Al rights reserved under Pan American and International Copyright Conventions. This Dover edition, First published in 1992, is an unabridged, corrected republica- tion of the work first published in two volumes by the Addison-Wesley Publishing Company, Reading, Mass., 1969 (Vol. One) and 1970 (Vol. Two). It was originally published in the “Addison-Wesley Series in Advanced Physics.” Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. L150L Library of Congress Cataloging-in-Publication Data Byron, Frederick W. Mathematics of classical and quantum physics / Frederick W. Byron, Jz., Robert W. Fuller em. “Unabridged, corrected republication of the work first published in two volumes by the Addison-Wesley Publishing Company, Reading, Mass., 1969 (Vol. One) and 1970 (Vol. Two) ... . in the “Addison-Wesley series in advanced physics” —Top. verso. Includes bibliographical references and index. ISBN 0-486-67164-X (pbk.) 1. Mathematical physics. 2, Quantum theory. 1. Fuller, Robert W. II. Títle. QC20.B9 1992 530.1'5— de20 92-11943 ciP PREFACE This book is designed as a companion to the graduate level physics texts on classical mechanics, electricity, magnetism, and quantum mechanics. It grows out of a course given at Columbia University and taken by virtually all first year graduate students as a fourth basic course, thereby eliminating the need to cover this mathematical material in a piecemeal fashion within the physics courses, The two volumes into which the book is divided correspond roughly to the two semesters of the full-year course. The consolidation of the mathematics needed for graduate physics into a single course permits a unified treatment applicable to many branches of physics. At the same time the fragments of mathematical knowledge possesed by the student can be pulled together and organized in a way that is especially relevant to physics. The central unifying theme about which this book is organized is the concept of a vector space. To demonstrate the role of mathematics in physics, we have included numerous physical applications in the body of the text, as well as many problems of a physical nature. Although the book is designed as a textbook to complement the basic physics courses, it aims at something more than just equipping the physicist with the mathematical techniques he needs in courses. The mathematics used in physics has changed greatly in the last forty years. It is certain to change even more rapidiy during the working lifetime of physicists being educated today. Thus, the physicist must have an acquaintance with abstract mathematics if he is to keep up with his own field as the mathematical language in which it is expressed changes. Ki is one of the purposes of this book to introduce the physicist to the language and the style of mathematics as well as the content of those particular subjects. which have contemporary relevance in physics. The book is essentially self-contained, assuming only the standard under= graduate preparation in physics and mathematics; that is, intermediate mechanics, electricity and magnetism, introductory quantum mechanics, advanced caleulus and differential equations. The level of mathematical rigor is generally comparable to that typical of mathematical texts, but not uniformly so. The degree of rigor and abstraction varies with the subject. The topics treated are of varied subtlety and mathematical sophistication, and a logical completeness that is illuminating in one topic would be tedious in another, While it is certainly true that one does not need to be able to follow the proof of Weierstrass's theorem or the Cauchy-Goursat theorem in order to be able to v vi PREFACE compute Fourier coefficients or perform residue integrals, we feel that the student who has studied these proofs will stand a better chance of growing mathematically after his formal coursework has ended. No reference work, let alone a text, can cover all the mathematical results that a student will need. What is perhaps possi- ble, is to generate in the student the confidence that he can find what he needs in the mathematical literature, and that he can understand it and use it. Tt is our aim to treat the limited number of subjects we do treat in enough detail so that after reading this book physics students will not hesitate to make direct use of the mathematical literature in their research. The backbone of the book-—the theory of vector spaces—is in Chapters 3, 4, and 5. Our presentation of this material has been greatly influenced by P. R. Halmos's text, Finite- Dimensional Vector Spaces. A peneration of theoretical physicists has learned its vector space theory from this book. Halmos's organiza- tion of the theory of vector spaces has become so second-nature that it is impossible to acknowledge adequately his influence. Chapters 1 and 2 are devoted primarily to the mathematics of classical physics. Chapter 1 is designed both as a review of well-known things and as an introduction of things to come. Vectors are treated in their familiar three-dimensional setting, while notation and terminology are introduced, preparing the way for subsequent generalization to abstract vectors in a vector space, In Chapter 2 we detour slightly in order to cover the mathematics of classical mechanics and develop the varia- tional concepts which we shall use later. Chapters 3 and 4 cover the theory of finite dimensional vector spaces and operators in a way that leads, without need for subsequent revision, to infinite dimensional vector spaces (Hilbert space)—the mathematical setting of quantum mechanics. Hilbert space, the subject of Chap- ter 5, also provides a very convenient and unifying framework for the discussion of many of the special functions of mathematical physics. Chapter 6 on analytic function theory marks an interlude in which we establish techniques and results that are required in all branches of mathematical physics. The theme of vector spaces is interrupted in this chapter, but the relevance to physics does not diminish. Then in Chapters 7, 8, and 9 we introduce the student to several of the most im- portant techniques of theoretical physics —the Green's function method of solving differential and partial differential equations and the theory of integral equations. Finally, in Chapter 10 we give an introduction to a subject of ever increasing im- portance in physics—the theory of groups. A special effort has been made to make the problems a useful adjunct to the text. We believe that only through a concerted attack on interesting problems can a student really “learn” any subject, so we have tried to provide a large selection of problems at the end of each chapter, some illustrating or extending mathematical points, others stressing physical applications of techniques developed in the text. In the later chapters of the book, some rather significant results are left as problems or even as a programmed series of problems, on the theory that as the student de- velops confidence and sophistication in the early chapters he will be able, with a few hints, to obtain some nontrivial results for himself. VOLUME ONE Vectors in Classical Physics CONTENTS Introduction. o 1 Geometric and Algebraio Definitions ofa | Vector 1 The Resolution of a Vector into Components 3 The Scalar Product , . Vo 4 Rotation of the Coordinate System: Orthogonal Transformations .. 5 The Vector Product, . . . . 14 A Vector Treatment of Classical Orbit Theory . . 17 Differential Operations on Scalar and Vector Fields “a 19 Cartesian-Tensors . |. “a 33 Calculus of Variations Introduction . .. Ca aa . 43 Some Famous Problems PR 43 The Euler-Lagrange Equation . 45 Some Famous Solutions 49 Isoperimetric Problems — Constraints 53 Application to Classical Mechanics . e Extremization of Multiple Integral... 6s Invariance Principles and Noether's Theorem Kva Vectors and Matrices Introduction . co . 85 Groups, Fields, and Vector Spaces . . 8 Linear Independence . Ls . 89 Bases and Dimensionality . 92 Isomorphisms . |. PR . 95 Linear Transformations. . . a 98 The Inverse of a Linear Transformation . 100 Matrices . Lc . 102 Determinants. a . 109 Similarity Transformations 117 viii 310 “3 CONTENTS Eigenvalues and Eigenvectors , The Kronecker Product, Vector Spaces in Physics Introduction. . cc a The Inner Product . Ca a a a a Orthogonality and Completeness . Complete Orthonormal Sets Self-Adjoint (Hermitian and Symmetric) Transformations Tsometries—Unitary and Orthogonal Transformations . The Eigenvalues and Eigenvectors of Self-Adjoint and Isometric Transformations . Diagonalization . On the Solvability of Linear Equations Minimum Principles. Ce Normal Modes . .. Cr a Perturbation Theory —Nondegenerate Case . PR Perturbation Theory —Degenerate Case . Hilbert Spaco— Complete Orthonormal! Sets of Functions Introduction , Function Space and Hilbert Space Complete Orthonormal Sets of Functions The Dirac 8-Function Weierstrass's Theorem: Approximation by Polynomiais Legendre Polynomials PP Fourier Series Fourier Integrals. . Spherical Harmonics and Associated Legendre Functions . Hermite Polynomials Sturm-Liouville Systems—Orthogonal Polynomiais A Mathematical Formulation of Quantum Mechanics . VOLUME TWO Elements and Applications of the Theory of Analytic Functions Introduction . Analytic Functions-— The Cauchy Riemann Conditions Some Basic Analytic Functions . Complex Integration—The Cauchy-Goursat Theorem . Consequences of Cauchy's Theorem Hilbert Transforms and the Canchy Principal Value An Introduction to Dispersion Relations . The Expansion of an Analytic Function in a Power Series - Residue Theory —Evaluation of Real Definite Integrals and Summation of Series . Applications to Special Functions and Integral Representations . ix 10 130 142 142 145 148 151 156 158 164 In 175 184 192 198 212 213 27 224 228 233 239 246 253 261 263 21 305 306 312 322 330 335 340 349 358 3 VOLUME ONE CHAPTER | VECTORS IN CLASSICAL PHYSICS INTRODUCTION Tn this chapter we shall review informally the properties of the vectors and vector fields that occur in classical physics. But we shall do so in a way, and in a notation, that leads to the more abstract discussion of vectors in later chapters. The aim here is to bridge the gap between classical three-dimensional vector analysis and the formulation of abstract vector spaces, which is the mathematical language of quantum physics. Many of the ideas that will be developed more abstractly and thoroughly in later chapters will be anticipated in the familiar three-dimensional setting here. This should provide the sub- sequent treatment with more intuitive content. This chapter will also provide a brief recapitulation of classical physics, much of which can be elegantly stated in the language of vector analysis—which was, of course, devised ex- pressly for this purpose. Our purpose here is one of informal introduction and review; accordingly, the mathematical development will not be as rigorous as in subsequent chapters. Fig. 1.1 Three equivalent vectors in a two-dimensional space. 1.1 GEOMETRIC AND ALGEBRAIC DEFINITIONS OF A VECTOR In elementary physics courses the geometric aspect of vectors is emphasized. A vector, x, is first conceived as a directed fine segment, or a quantity with both a magnitude and a direction, such as a velocity or a force. A vector is thus distinguished from a scalar, a quantity which has only magnitude such as temperature, entropy, or mass. In the two-dimensional space depicted in Fig. 1.1, three vectors of equal magnitude and direction are shown. They form an 1 2 VECTORS IN CLASSICAL PHYSICS Li equivalence class which may be represented by Vo, the unique vector whose initial point is at the origin. We shall gradually replace this elementary characterization of vectors and scalars with a more fundamental one. But first we must develop another language with which to discuss vectors. An algebraic aspect of a vector is suggested by the one-to-one correspond- ence between the unique vectors (issuing from the origin) that represent equiva- lence classes of vectors, and the coordinates of their terminal points, the ordered pairs of real numbers (x, x). Similarly, in three-dimensional space we associate a geometrical vector with an ordered triple of real numbers, (x, x,, x), which are called the components of the vector. We may write this vector more briefiy as x; where it is understood that i extends from 1 to 3. In spaces of dimen- sion greater than three we rely increasingly on the algebraic notion of a vector, as an ordered n-tuple of real numbers, (x, x; ***, x,). But even though we can no longer construct physical vectors for n greater than three, we retain the geometrical language for these n-dimensional generalizations. A formal treat- ment of the properties of such abstract vectors, which are important in the theory of relativity and quantum mechanics, will be the subject of Chapters 3 and 4, In this chapter we shall restrict our attention to the three-dimensional case. There are then these two complementary aspects of a vector: the geometric, or physical, and the algebraic. These correspond to plane (or solid) geometry and analytic geometry. The geometric aspect was discovered first and stood alone for centuries until Descartes discovered algebraic or analytic geometry. Anything that can be proved geometrically can be proved algebraically and vice- versa, but the proof of a given proposition may be far easier in one language than in the other. Thus the algebraic language is more than a simple alternative to the geo- metric language. Tt allows us to formulate certain questions more easily than we could in the geometric language. For example, the tangent to a curve at a point can be defined very simply in the algebraic language, thus facilitating further study of the whole range of problems surrounding this important con- cept. Itis from just this formulation of the problem of tangents that the cal- culus arose. It is said of Niels Bohr that he never felt he understood philosophical ideas until he had discussed them with himself in German, French, and English as well as in his native Danish. Similarly, one's understanding of geometry is strengthened when ong can view the basic theorems from both the geometric and the algebraic points of view. The same is true of the study of vectors. It is all too easy to rely on the algebraic language to carry one through vector analysis, skipping blithely over the physical, geometric interpretation of the differential operators. We shall try to bring out the physical meanings of these operators as well as review their algebraic manipulation. The basic operators of vector analysis crop up everywhere in physics, so it pays to develop a physi- cal picture of what these operators do—that is, what features they “measure” of the scalar or vector fields on which they operate. 4 VECTORS IN CLASSICAL PHYSICS 13 1.3 THE SCALAR PRODUCT The scalar (“inner” or “dot” product of two vectors x and y is the real number defined in geometrical language by the equation xy = |x| |y|cos8, where É is the angle between the two vectors, measured from x to y. Since cos 6 is an even function, the scalar product is commutative: Wy=yx. Moreover, the scalar product is distributive with respect to addition: xy+ty=xy+xz. This equation has a familiar and reasonable appearance, but that is only be- cause we automatically interpret it algebraically, where we usually take distri- butivity for granted, The reader will find it instructive to prove this by geo- metrical construction. If x-y = 0, it does not follow that one or both of the vectors are zero. It may be that they are perpendicular. Note that the length of a vector x is given by Ixl=*= (x, since cos9 = 1 for 8 = 0. In particular, for Cartesian basis vectors, we have ee =ô. (1.1) where 8, is the Kronecker delta defined by Lodfoi= 1) = ( . 12 “Olo car dx “2 1f we expand two arbitrary vectors, x and y, in terms of the Cartesian basis, 3 x= xe + xe, + xe = Do xe , fm y = e + ve + es = Sue, , then o xy=(D xe) (Dye) f 7 = 2 Xyere= » x); 03; = (1.3) Here we have used the distributivity of the scalar product; > stands for x õ . TT EE) 1.4 ROTATION OF THE COORDINATE SYSTEM 5 This last expression may be taken as the algebraic definition of the scalar product. Tt follows that the length of a vector is given in terms of the scalar product by lx) = (xx)? = (5,x))!2. This equation provides an independent way of as- sociating with any vector, a number called its length. We see that the notion of length need not be taken as inherent in the notion of vector, but is rather a consequence of defininp a scalar product in a space of abstract vectors. Thus in Chapter 3 we shall study abstract vector spaces in which no notion of length has been defined, Then, in Chapter 4 we shall add an inner (or scalar) product to this vector space and focus on the enriched structure that results from this addition. We shall now introduce a notational shorthand known as the Einstein sumimation convention. Einstein, in working with vectors and tensors, noticed that whenever there was a summation over a given subscript (or superscript), that subscript appeared iwice in the summed expression, and vice versa. Thus one could simply omit the redundant summation signs, interpreting an expres- sion like x,); to mean summation over the repeated subscript from 1 to, in our case, 3. If there are two distinct repeated subscripts, two summations are im- plied, and so on. Ina letter, Einstein refers with tongue in cheek to this obser- vation as “a great discovery in mathematics,” but if you don't believe it is, just try getting along without it! (Another story in this connection— probably apocryphal—has it that the printer who was setting type for one of Einstein's papers noticed the redundancy and suggested omitting the summation signs.) We shall adopt Einstein's summation convention throughout this chapter. Tn terms of this convention we have, for example, X=— Xe, XY =)0 =); Dio X€4=Xe'€=X0, =X. The last equation defines x;, the component of x in the e, direction, x-e; is also caíled the projection of x on the e; axis. The set of numbers (x;) is called the representation (or the coordinates) of the vector x in the basis (or the coordinate system) (e). 1.4 ROTATION OF THE COORDINATE SYSTEM; ORTHOGONAL TRANSFORMATIONS We shall now consider the relationship between the components of a vector expressed with respect to two different Cartesian bases with the same origin, as shown in Fig. 1.3. Any vector x can be resolved into components with respect to either the K or the K' system. For example, in K we have x= (xeje;= x;, (1.4) 1.4 ROTATION OF THE COORDINATE SYSTEM 7 Equation (1,8) stands for a set of nine equations (of which only six are distinct), each involving a sum of three quadratic terms. It is left to the reader to show (by expanding the unprimed vectors in terms of the primed basis and taking scalar products) that we also have the relation Gu = O. (1.9) The expressions (1.8) and (1.9) are referred to as orthogonality relations; the corresponding transformations (Eg. 1.5) are called orthogona! transformations. In an n-dimensional space, the rotation matrix will have nº elements, upon which the orthogonality relations place 4(xº + n) conditions, as the reader can verify. Thus Po Hr+n=inn—1) of the ay; are left undetermined. In a two-dimensional space this leaves one free parameter, which we may take as the angle of rotation. In a three-dimen- sional space there are three degrees of freedom, corresponding to the three so- called Euler angles used to describe the orientation of a rigid body. Equation (1.5), together with the orthogonal!ity relations, tells us how one set of orthogonal basis vectors is expressed in terms of another rotated set. Now weask: How are the components of a vector in K related to the components of that vectors in K”, and vice versa? Any vector x may be expressed either in the K system as x = x,e; or in the K” system as x = x/e!. Let us first express the x;, the components of x with respect to the basis e,, in terms of the x/, the components of x with respect to the basis e!. Using Eg. (1.5), we have X = Xe] = XiQye, = Xe. (1.10) Now, since the basis vectors are orthogonal, we may identify their coefficients in Eg. (1.10): x — ax. (1.11) (More Formally, if are, = fre, then (a; — B)e; = 0, Since this is a sum, it does not follow automatically that «, = 8;. However, taking the scalar product with e gives (a, — 80 = 0, whence q, = 8;.) To derive the inverse transformation, one could, of course, repeat the above procedure, substituting for the unprimed vectors. That is, instead of using Eq. (1.5), one could use the corresponding relation for the unprimed basis vectors in terms of the primed: e = (eee =aje;. However, using the orthogonality relations, we can derive this result directly from Eg. (1.11). Multiplying it by ax, summing over j, and using Eq. (1.8), we have = 1 = aux; = Awtiyx) = Gux) = Xks (1.12) which gives the primed comporenis in terms of the unprimed components, 8 VECTORS IN CLASSICAL PHYSICS L4 In summary, we have x= xe =x, !— / E = Ge; e 18) (1.13) =x! 4X; X — TX Au; — Qua = dj. K should be understood that these equations refer to the components of one vector, x, as expressed with respect to two diflerent sets of basis vectors, e; and e;. Thus unprimed basis vectors can be expressed in terms of the other (primed) basis. Thus a, is the jth component of ef, expressed with respect to the unprimed basis, and a; is the jth component of e; expressed with respect to the primed basis, x 2 Fig. 1.4 Rotation in two dimensions, or ' x a - à es jo rotation in three dimensions about an e axis, xs, orthogonal to the x1, x», x, x! xr axes. Example 1.1, The iwo-dimensiona! rotation matrix. We have defined the ele- ments of the rotation matrix in Egs. (1.5) and (1.6). For the two-dimensional case we have four coefficients: a; = (ee), fori,j= 1,2. From Fig. 1.4 àt is clear that cos sin (a) = [ os pp 9]. (1.149) -siny cos q. The first subscript of a; labels the row and the second subscript labels the column of the element a,;. This rotation matrix tells us what happens to the components of a single vector x when we go from one basis, e, to a new basis, ei, by rotating the basis counterclockwise through an angle (+). From Eq. (1.13), the components x, of a vector x relative to the e; basis and the compo- nents x; of that same vector relative to the e; basis are related by x; = a;x; or written out in full, x = cosgx + sinyx;, , . (1.15a) x = —sin px, + Cos px. Here x, and x/ refer to the components of a single vector with respect to two bases. The vector x sits passively as the basis with respect to which it is expressed rotates beneath it. 10 — VECTORS IN CLASSICAL PHYSICS 1.4 Example 1.2. The three-dimensional rotation matrix R(y, 8,4). Suppose that we want to transform to a coordinate system in which the new z-axis, xi, is in an arbitrarily specified direction, say along the vector V in Fig. 1.5. Such a rotation may be compounded of two three-dimensional rotations about an axis, such as those discussed in Example 1.1. First we rotate the coordinate system counterclockwise about the common x;-x) axis through an angle q. This gives eia, (1.18) where the a; are given by Eg. (1.17). Now we rotate clockwise through an angie 8, that is, counterciockwise through the angle (—6), in the x$x/-plane about the xi-axis. (We could as well have rotated about the x/ axis but the sequence we have chosen is the conventional one.) The appropriate rotation matrix for this rotation of base about the xi-axis is cos 8 0 —sin 8 (bi) o 1 o |, (1.19) sin é [o cos 8 and the new basis vectors, e”, are given in terms of the primed ones by e = bye. Fig. 1.5 The vector, V, which determines the z-axis of a rotated coordinate system. x Therefore, using Eq. (1.18), we have e! = bytes (1.20) To go directiy from the unprimed system to the doubly primed system we must know the cosfficients c; in the equation e = cum (1.21) Knowing these coefficients is equivalent to knowing the three-dimensional rota- tion matrix. From Egs. (1.20) and (1.21), we see that Cu = byap. (1.22) i4 ROTATION OF THE COORDINATE SYSTEM n Using this result and the matrices (1.17) and (1.19), we may compute the ele- ments cy. The resulting rotation matrix is cospcos8 sinqcosd —sin9 Re ==) sing cos q o |. (123) cos q sin É sin gsinê cos é This rotation matrix contains the matrix R(p) [Eq. (1.17)] as the special case 8 =0; thus R(p,0) = R(g). Equation (1.22) is a special case as the general operation of matrix multiplication which will be treated fully in Chapter 3. The components of a vector x relative to the e-basis and the components of that same vector x relative to the ef'-basis are related according to x! = e. (1.24) The rotation matrix R(y, 8) does not represent the most generai possible rotation. One more rotation is possible, a counterclockwise rotation through an angle & in the xi'x%-plane about the x5-axis. This third rotation about an axis is described by the rotation matrix cos sin & o (dj) =)-sinb cosh O jo) 0 1 And the grand “rotation of rotations,” that may be achieved by compounding the three rotations about axes, through q, é, and &, is described by the rotation matrix whose elements are ERtp, 9, PI; = dabuai] “The reader may verify that the resulting matrix is Rlp, 8,4) = cospeosfcosp —sinpsiny singcosócosf +cospsing —sing cosy —cospcosósiny —sinpcosy —sinpcosôsiny +cospcosy sinfsinç|. cos q sin 6 singsinô cosê (1.25) The angles (y, 8, 4) are called the Euler angles, Their definition varies widely — the probability is small that two distinct authors” general rotation matrix will be the same. Note that R(p,68,0) = R(y,0) and R(y,0,0) = K(g). The reader might note in passing that the determinant of R(p, 6, ) (and alí the other ro- tation matrices), has the value one. We shall prove this in Chapter 4. Originally we introduced a vector as an ordered triple of numbers. The rule for expressing the components of a vector in one coordinate system in terms of its components in another system tells us that if we fix our attention on a physical vector and then rotate the coordinate system (K—K”), the vector will have different numerical components in the rotated coordinate system. So we are led to realize that a vector is really more than an ordered triple. Rather, 
