Mathematics for Quantum Mechanics: An Introductory Survey of Operators, Eigenvalues, and Linear Vector Spaces 0486453081, 9780486453088

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MatheInatics for QuantUIn Mechanics An Introductory Sur"e..y of Operators, Eigen"alues, and Linear Vector Spaces

John David Jackson University of Illinois

w. A. Benjamin, Inc.

New York

1962

MATHEMATICS FOR QUANTUM MECHANICS An Introductory Survey Copyright@ 1962 by W. A. Benjamin, Inc. All rights reserved. Library of Congress Catalog Card Number: 62-17526 Manufactured in the United States of America

The manuscript was received April 1, 1962, and published July 20, 1962.

W. A. BENJAMIN, INC.

Editor's Foreword Everyone concerned with the teaching of physics at the advanced undergraduate or graduate level is aware of the continuing need for a modernization and reorganization of the basic course material. Despite the existence today of many good textbooks in these areas, there is always an appreciable time-lag in the incorporation of new view-points and techniques which result from the most recent developments in physics research. Typically these changes in concepts and material take place first in the personal lecture notes of some of those who teach graduate courses. Eventually, printed notes may appear, and some fraction of such notes evolve into textbooks or monographs. But much of this fresh material remains available only to a very limited audience, to the detriment of all. Our series aims at filling this gap in the literature of physics by presenting occasional volumes with a contemporary approach to the classical topics of physics at the advanced undergraduate and graduate level. Clarity and soundness of treatment will, we hope, mark these volumes, as well as the freshness of the approach. Another area in which the series hopes to make a contribution is by presenting useful supplementing material of well-defined scope. This may take the form of a survey of relevant mathematical principles, or a collection of reprints of basic papers in a field. Here the aim is to provide the instructor with added flexibility through the use of supplements at relatively low cost. The scope of both the lecture notes and supplements is somewhat different from the "Frontiers in Physics" series. In spite of wide variations from institution to institution as to what comprises the basic graduate course program, there is a widely accepted group of "bread and butter" courses that deal with the classic topics In physh::s. ThE)SC include: Mathematical methods of physics,

vi

EDITORS' FOREWORD

electromagnetic theory, advanced dynamics, quantum mechanics, statistical mechanics, and frequently nuclear physics and/or solid state physics. It is chiefly these areas that will be covered by the present series. The listing is perhaps best described as including all advanced undergraduate and graduate courses which are at a level below seminar courses dealing entirely with current research topics. The publishing format for the series is in keeping with its intentions. Photo-offset printing is used throughout, and the books are paperbound in order to speed publication and reduce costs. It is hoped that books will thereby be within the financial reach of graduate students in this country and abroad. Finally, because the series represents something of an experiment on the part of the editors and the publisher, suggestings from interested readers as to format, contributors, and contributions will be most welcome. J. DAVID JACKSON DAVID PINES

Preface The purpose of these notes is to present concisely the mathematical methods of quantum mechanics in a form which emphasizes the unity of the different techniques. Since the methods are applicable to the description of many physical systems outside the domain of quantum theory, the material may be useful in other areas. But the orientation is toward the graduate or advanced undergraduate student beginning a serious study of quantum mechanics. The notes were developed as a supplement for the first-year graduate course in quantum mechanics at the University of Illinois. At all but a few graduate schools in physics, the entering students come with a variety of mathematical backgrounds, ranging from ignorance of Fourier series and partial differential equations, on the one hand, to familiarity with group theory and Banach spaces, on the other. The teaching of quantum mechanics to such a heterogeneous group presents problems. It was in an attempt to solve some of these pr.oblems that the present little volume came into being. My aim was to assure that everyone had a certain level of knowledge in those areas of mathematics that bear most directly on quantum mechanics. The level is not high, to be sure, but it is adequate for the beginning student. When teaching quantum mechanics, I personally spend five or six weeks at the start in covering the material presented here. Then I feel free to discuss the physical subject with whatever formalism is most appropriate for the topic at hand. But others may wish to discuss the relevant mathematics as the need .. arises, or even assume that the student can learn it outside the lecture room. Whatever the attitude, I hope that these notes will serve both teacher and student by bringing together in compact form tpe essential mathematical background for quantum mechanics. Urbana, Illinois April 15, 1962

J. D. JACKSON vi i

Contents Editors' Foreword Preface

v vii

1 Introductory Remarks References 2 Eigenvalue Problems in Classical Physics

1 2 4

2-1

Vibrating string

4

2-2

Vibrating circular membrane

6

2- 3 Small oscillations of a mechanical system

10

2- 4

15

Rotation of axes and orthogonal transformations

2- 5 Euler's theorem and principal-axes transformations as eigenvalue problems

3 Orthogonal Functions and Expansions

18 )2

3-1

Fourier series

22

3-2

Expansion in orthonormal functions

25

3- 3

Dirac delta function and closure relation

28

3- 4

Bessel functions as an orthonormal set on the interval (0,1)

31

3- 5

Schmidt orthogonalization method

33

3-6

Legendre polynomials

35

CONTENTS

x 3-7

Other orthogonal polynomials

36

3- 8

Fourier integrals

37

4 Sturm-Liouville Theory and Linear Operators on Functions

41

4-1

Sturm-Liouville eigenvalue problem

41

4-2

Linear operators on functions

43

4- 3

Eigenvalue problem for a linear Hermitean operator

45

Further properties of operators

46

4-4

5 Linear Vector Spaces

48

5-1

State vectors and representatives

48

5-2

Complex vectors in a n-dimensional Euclidean space

49

5-3

Basis and base vectors

51

5-4

Change of basis

53

5- 5

Linear operators and their matrix representation

56

5-6

Further definitions and properties of linear operators

59

5-7

Unitary operators and equations of motion

63

5-8

Eigenvectors, eigenvalues, and spectral representation

66

Determination of eigenvalues and eigenvectors

68

5-9

5-10 Transition to Hilbert space; Dirac notation

72

5-11 State vectors and wave functions

75

Appendix A: Bessel (Cylinder) Functions

79

Appendix B: Legendre Functions and Spherical Harmonics

88

Mathematics for

Quantum Mechanics

1 Introductory Remarks The purpose of these notes is to set forth the essentials of the mathematics of quantum mechanics with only enough mathematical rigor to avoid mistakes in the physical applications. In various parts of quantum theory it is appropriate to use mathematical methods that at first sight are quite different and unconnected. Thus in dealing with potential barriers or the hydrogen atom, the techniques of ordinary or partial differential equations in coordinate space are employed, whereas for a problem such as the harmonic oscillator, the use of abstract linear operator methods leads to an elegant solution. The chief aims of the present discussion are to show the underlying unity of all the methods and to build up enough familiarity with each of them so that in the subsequent treatment of quantum mechanics as a subject of physics the best method can be applied to each problem without apology and without the need to explain new mathematics. Quantum theory was originally developed with two different mathematical techniques- Schrodinger wave mechanics (differential equations) and Heisenberg matrix mechanics. The equivalence of the two approaches was soon demonstrated, and the mathematical methods were generalized by Dirac, who showed that the techniques of Heisenberg and Schrodinger were special representations of the formalism of linear operators in an abstract vector space. The concept of discreteness is central in quantum theory. Physically measurable quantities (called "observables") are often found to take on only certain definite values, independent of external conditions (e.g., preparation of light source, detailed design of deflecting magnet, etc.). Important examples of discrete observables are energy (Ritz combination principle and Rydberg formula, Franck-Hertz oxpertnlent) and anp;ular momenlum (Stern-Gerlach experiment). In

2

MATHEMATICS FOR QUANTUM MECHANICS

mathematical language the discrete, allowed values of an observable are called eigenvalues (sometimes called characteristic or proper values). The physicist is often interested in predicting and correlating the eigenvalues for a given physical system. Provided he has an appropriate mathematical description of the physical system, he wants to solve "the eigenvalue problem." Thus the mathematical eigenvalue problem is an important aspect of quantum mechanics. This problem can be phrased in terms of differential equations, in terms of matrices, or in terms of linear vector spaces. We shall consider the various techniques and explore their essential unity below. Not all of quantum mechanics concerns itself with discrete eigenvalues, of course. Hence some of the mathematical discussion will not bear directly on the eigenvalue problem. Furthermore, a number of topics, such as perturbation theory and variational methods, will be omitted from these notes, to be dealt with separately.

References Since only the bare essentials will be presented in these notes, the student will wish to consult more complete treatments. Some useful references are the following: R. Courant and D. Hilbert, "Methods of Mathematical Physics," Vol. I, Interscience, New York, 1953. Chapter 2 on orthogonal expansions; Chap. 5 on eigenvalue problems; Chap. 7 on special functions. B. Friedman, "Principles and Techniques of Applied Mathematics," Wiley, New York, 1956. A very useful, if formal, treatment. G. Goertzel and N. Tralli, "Some Mathematical Methods of Physics," McGraw-Hill, New York, 1960. P. R. Halmos, "Finite Dimensional Vector Spaces," Princeton University Press, Princeton, N.J., 1942; "Introduction to Hilbert Space," Chelsea, New York, 1~51. These books present a rigorous mathematical discussion of vector spaces. F. B. Hildebrand, "Methods of Applied Mathematics," Prentice-Hall, Englewood Cliffs, N.J., 1952. The first 100 pages deal with matrices and vector spaces. H. Margenau and G. M. Murphy, "Mathematics of Physics and Chemistry," 2nd ed., Van Nostrand, Princeton, N.J., 1956. Chapters 2, 3, 7, 8, 10, and 11 have particular relevance. P. M. Morse and H. Feshbach, "Methods of Theoretical Physics," 2 vols., McGraw-Hill, New York, 1953. Very complete, with valuable appendices at the end of each chapter. V. Rojansky, "Introductory Quantum Mechanics," Prentice-Hall, Englewood Cliffs, N.J., 1946. Chapter IX has a useful review of matrices. Parts of Chaps. X and XI are also relevant.

INTRODUCTORY REMARKS

3

H. Sagan, "Boundary and Eigenvalue Problems in Mathematical Physics," Wiley, New York, 1961. A. Sommerfeld, "Partial Differential Equations," Academic, New York, 1949. The emphasis is on orthonormal expansions, special functions, eigenfunctions, and eigenvalues. Special mention must be made of one extremely useful reference on special functions;

w. Magnus

and F. Oberhettinger, "Formulas and Theorems for the Special Functions of Mathematical Physics," Chelsea, New York, 1949. This book will be referred to often and will be denoted as "MO."

A much more elaborate collection of information on special functions and various transforms is contained in the Bateman volumes, Bateman Manuscript Project, "Higher Transcendental Functions," 3 vols., A. Erdelyi (Ed.), McGraw-Hill, New York, 1953. Bateman Manuscript Project, "Tables of Integral Transforms," 2 vols., A. Erdelyi (Ed.), McGraw-Hill, New York, 1954. Note, however, that MO has quite useful tables of Fourier and Laplace transforms. When it comes to tables of integrals and numerical values of the elementary functions, personal preference takes over. Useful, inexpensive references include the following: H. B. Dwight, "Tables of Integrals and Other Mathematical Data," Macmillan, New York. B. O. Pierce and R. M. Foster, "A Short Table of Integrals," 4th ed. Ginn, Boston. Handboo.k of Chemistry and Physics, "Mathematical Tables," McGraw-Hill, New York. E. Jahnke and F. Emde, "Tables of Functions" (paperback), Dover, New York, 1945. Tabulation and graphs of special functions. As a final introductory remark let me say that it is assumed that the student has some familiarity with ordinary differential equations, the method of separation of variables for partial differential equa- • tions, the elements of Fourier series, the elementary properties of matrices, and the notions of vectors in three dimensions, rotations, etc. Furthermore, it is assumed that his knowledge of classical mechanics is at the level of the books by Symon (K. R. Symon, "Mechanics," 2nd ed., Addison-Wesley, Reading, Mass., 1960) or by Slater and Frank (J. C. Slater and N. H. Frank, "Mechanics," McGraw-Hill, New York, 1947).

2 Eigenvalue Problems in Classical Physics Eigenvalue problems dominate quantum mechanics, but they were well known and understood in classical physics. They occur for mechanical systems with a finite number of degrees of freedom (discrete systems), or for mechanical or electromagnetic systems with an infinite number of degrees of freedom (continuous systems). We shall discuss briefly a few examples to recall the mathematical methods used, and to see how and why eigenvalues arise.

2-1

VIBRATING STRING

A string of uniform mass density p, tension T, fastened at the points x = 0 and x = a, can move in the xy plane with a displacement measured perpendicular to the x axis equal to u(x, t). The equation of motion for small oscillations of the string is a linear, second-order, partial differential equation of the form

If we define a quantity with dimensions of velocity v = (T/p)1/2, the equation can be written 2

a u2 _ 1- a\t= 0 ax

(2)

y2 at2

Using the separation-of-variables technique, we assume a solution u(x, t)

= y(x)z (t)

(3)

and find that y(x) and z(t) must satisfy the separate ordinary differential equations 4

EIGENVALUE

5

PROBLEMS

(4)

where k 2 is some as yet undetermined constant. We note that y(x) and z(vt) satisfy the same equation whose solution is sines and cosines. So far there has been no mention of an eigenvalue problem. That comes with the imposition of the boundary conditions. These are 1. Spatial: u(O,t) = u(a,t) = 0, for all t. 2. Temporal: u(x,O) = F(x), au/at (x,O) = G(x), for ~ x ~ a. ., First we consider the spatial boundary condition. The solution for y(x) is

°

y(x) = A sin kx + B cos kx To satisfy boundary condition 1 it is necessary that y(O) = y(a) = 0. Hence B = and, if A =1= 0, then ka = n 1T. The unknown parameter k has thus been found to have a countably infinite set of discrete eigenvalues,

°

k = n1T n a

(n = 1,2,3, .. .)

The most general solution consistent with the spatial boundary conditions is, therefore, 00 • n 1Tx [ An cos n 1Tvt + B SIn . n 1T.-vt ] SIn u ( x, t ) = 'E n n= 1 a a a

(7)

Note that linear superposition has been assumed valid, consistent with the iinearity of the original differential equation. The coefficients An and Bn are determined by the temporal boundary conditions at t = 0. Since this is a problem in Fourier series inversion which will be discussed below (Sec. 3-1), we shall not bother with it here. The essential point is the discrete nature of the parameter k as a result of the spatial boundary conditions. A direct consequence of this discreteness is the discreteness of the frequencies of vibration . of the string. If the fundamental frequency is defined as w = 1Tv/a, the allowed frequencies of vibration are W

n

= nw

(n = 1,2,3, ... )

The mechanical motions associated with these frequencies [y = sin(wnx Iv) J are called the normal modes or eigenmodes of

(8)

6

MATHEMATICS FOR QUANTUM MECHANICS

oscillation. An arbitrary small-amplitude motion of the string can be built up by linear superposition, according to (7).

Problem: Show how eigenfrequencies of oscillation occur for the small oscillations of a uniform, flexible, rectangular membrane with sides of length a and b. Find the most general solution for the displacement, and show that the eigenfrequencies can be written

where m and n are positive integers. 2-2

VIBRATING CIRCULAR MEMBRANE

The problem of the eigenfrequencies of a circular membrane, besides involving an additional independent variable, demands explicit consideration of another physical condition on the solution besides the satisfying of boundary conditions. That physical condition is the single-valuedness of the displacement. We shall see that this requirement, as much as the boundary conditions, determines the eigenvalues of the problem. The small oscillations of an elastic membrane are described by the partial differential equation (9)

where u(x,y,t) is the displacement and v is a velocity characteristic of the membrane. Because the boundary condition of zero displacement is to be applied along a circle, it is obviously convenient to use polar coordinates (p,cp) instead qf (x,y). The two-dimensional Laplacian in polar coordinates can be found in MO, p. 145.t The transformed equation is

o

(10)

where u is now u(p,cp,t). Again separating variables by writing u = R(p)cll(cp)z(t), we find that each of the functions R,cll,z satisfy ordinary differential equations

t MO stands for the book by Magnus and Oberhettinger cited in the reference list in Chapter 1. Other books in the reference list will be' cited in the'text by the names of the authors.

EIGENVALUE

PROBLEMS

7

(11) d 2R ( 2 + 1 dR ~p2 dp + k

P

2 -

1I )

p2 R

=

0

In these equations, k 2 and 1I2 are as yet undetermined separation constants. If k 2 > 0, the time dependence will be oscillatory, as desired. To specify an eigenvalue problem we must prescribe boundary, and/or other, conditions on the solution u(p,cp,t). The spatial boundary condition is u = 0 for p = a, independent of cp and t. Hence R(a)

=0

(12)

But the other spatial variable cp is limited in its physical range (0 ~ cp ~ 21T), and, for a circular membrane, has no role in the boundary conditions. Instead, the physical requirement of single-valuedness of u(p,cp,t) inside p = a makes the condition

u(p, cp, t)

= u(p,cp + 21T p, t)

where p is any integer. The solution for no the mean-square error Mn can be made smaller than any arbitrarily chosen small positive number. Then the series representation, 00

f(x) ==

~

(19)

a cp (x) j==1 J J o

0

with coefficients (15), is said to converge in the mean to f(x) on the interval (a,b). The equality in (19) is to be interpreted as meaning that the series on the right-hand side is equal to f(x) almost everywhere on the interval (a, b) The problem of proving that an orthonormal set is complete is not trivial in general. Sometimes, as for the sine and cosine expansions of Fourier series, or for orthonormal polynomials such as Legendre polynomials, the completeness can be interpreted as a consequence of the completeness of the powers zn in a Laurent expansion. Orthonormal sets of eigenfunctions can also be shown to be complete (Courant and Hilbert, p. 368, Margenau and Murphy, p. 277, Morse and Feshbach, p. 738). We shall take for granted the completeness of any sets of functions which arise in physical applications. 0

Problem: (a) Show that the functions e imx , with m == ( ... ,-2,-1,Q,1,2, ...) can be used to form an orthnormal set CPm (x) on the interval (O,21T). (b) Determine explicitly the expansion coefficients am and relate them to the Fourier series (1) and (3). 3-3

DIRAC DELTA FUNCTION AND CLOSURE RELATION'

A special "function" of great importance in mathematical physics is the Dirac delta function 6(x - a). The word function was put in quotation marks in the preceding sentence because the Dirac delta function is not at all well behaved in a mathematical sense. It is de- ' fined by the following statements:

ORTHOGONAL FUNCTIONS for x

(1) a(x - a) == 0

(2)

f

29 =1=

a

c 6(x - a) dx

=1

if a is contained in the inte rval (b, c)

== 0

if a is not in the interval (b,c)

b

(20)

(If a is an end point of the interval (b,c), the integral is usually defined to be equal to !.) 'Evidently the Dirac delta function can be defined as the limiting form of a sharply peaked curve which becomes higher and higher but narrower and narrower, so that the area under curve is maintained constant. For example, a Gaussian can be used. Then

It is a straightforward matter to establish the following properties of integrals involving delta functions: Let f(x) be an arbitrary function defined in the neighborhood of x == a. Let the interval of integration include x == a. (l) !f(X)/j(X - a) dx

~ f(a)

(21)

(2) If o(n)(x - a) means the nth derivative of o(x - a) with . respect to its argument,

f f(X)6 (0) (x -

f

a) dx

f(x)6(y(x» dx

=

= (- 1)of(0) (a)

t

(22)

f(xi)

I y'(xi) I

(23)

where the points xi are the real roots of y(x) == 0 in the interval of integration. Equation (23) is equivalent to the statement o(x - xi)

6(y(x»

= ~ I y'(Xi) I

(24)

1

and follows fronl o(y) dy = o(x) dx. The delta function can be defined in more than one dimension. In three dimensions, for example, if positions are described by coordinate vectors x with components Xt,X2,X3, then (25)

30

MATHEMATICS FOR QUANTUM MECHANICS

The integrals in (20), (21), (22), and (23) now become three-dimensional integrals in an obvious way. Sometimes in three dimensions it is convenient to use coordinates (~1'~2,~3) other than rectangular. Then the delta function (25) must be transformed according to the rule

This means that

where J(xi'~i) is the Jacobian of coordinate transformation. In spherical coordinates (r,8,cp), for example, we find 1 6(x - x') = J 6(r - r')o(cos 8 - cos e/)6(cp - cp') r

(27)

Now we return to the discussion of orthonormal expansions. Let us write (19) with the coefficients (15) inserted explicitly: f(x) = ~ (cp. ,f)cp. (x) J

j

or

J

b

f(x)

=

{~cp. (x)cp~ (x')

f

a

j

J

f(x / ) dx / }

(28)

J

In writing (28) we interchanged the summation and integral signs. If (28) is to hold for an arbitrary function f(x), the expression in braces under the integral must have a peculiar property. It must be such that, when multiplied by f(x') and integrated over the interval (a ~ x' ~ b), it yields the value of f at the point x. Comparison of (28) with (21) shows that the term in braces must be a delta function':

L: cp. (x)cp~ (x /)= 6(x .

J

J

J

x')

(29)

This relation is called the closure or completeness relation, and holds for any complete set of orthonormal functions. Note that (29) holds only if x and x' are on the interval (a,b) over which the set CPj (x) is orthogonal. The closure relation is sometimes called the expansion theorem, although this name is more appropriate in the context of linear vectoJ; spaces (see Sec. 5-3). The name "expansion theorem" stems from the following theorem:

ORTHOGONAL FUNCTIONS

31

If f and g are arbitrary functions, and the cpo are a complete set, then J

(f,g)

= ~ (f,cpj)(cpj,g) J

The theorem is proved simply from the definition of the inner product and the completeness relation (29), or from the expansion (19). This means that in a special sense we may write

L; j

cp.cp~

J J

=1

The interpretation of (31) is that on the left side of (30) we may insert unity between f and g in the inner product. Then by using (31) we may convert the innter product (f,g) into the sum on the right-hand side of (30). In quantum theory, the process of using (31) to obtain relation' (30) is called" the insertion of a complete set of states." It is often quite useful in practice when the inner products (cpj ,g) or (cpj ,f) are easier to evaluate or estimate than (f,g). 3-4

BESSEL FUNCTIONS AS AN ORTHONORMAL SET ON THE INTERVAL (0,1)

As a concrete, nontrivial example of orthonormal expansions, we consider the Bessel functions on the interval (0,1) in the independent variable z. In Sec. 2-2 we found that in solving a physical problem a series expansion in Bessel functions occurred. Thus we expect that Bessel functions form an orthonormal set on a finite interval. The series solution [Eq. (22), Chap. 2] for the displacement u(p,cp,t) of the circular membrane involves a double sum, one over m with sines and cos ines of (mcp), and one over n with the roots of J m (x) as the parameter. In view of the problem on p. 28, we expect that in the one variable z the set of functions m fixed (

)

n = 1,2,3, ...

will form an orthogonal set on the interval (0 ~ z ~ 1) for each value of m = 0,1,2, ... , separately. To show in what sense the Bessel functions are indeed orthogonal, it is necessary to look at the differential equation and boundary conditions satisfied by the Bessel functions. All the necessary results appear in Appendix A. In Eq. (16) of Appendix A we put II = Jl, Z ::: ~ == ;J. Then we have the indefinite integral

32

MATHEMATICS

FOR QUANTUM ME C HANICS

Z

(p2 _ q2)

f

Z' J

Jl

(pz')J (qz') dz' Jl

If the parameters p and q are chosen to be roots p

q

= x Jln

and

= xJln' of J Jl(x) = 0, then at z = 1 the right-hand side of (33) will

vanish. It can be shown with little difficulty that for Jl > -1, the right-hand side of (33) also vanishes at z = 0. Thus the orthogonality integral, 1

(xtn - xtn')

f

.

zJ /J.(x/J.nz)J /J.(x/J.n' z) dz

=0

(34)

o

is established. This shows that the functions and

(35)

are orthogonal on the interval (0,1), provided n' =1= n. [Sometimes this is stated in other ways, namely, that the Bessel functions are orthogonal on the interval (0,1), with a weighting factor z: or that the functions J Jl(X Jln Z l / 2) are orthogonal on the interval (0,1).] For n'

= n, equation (34) is satisfied by the vanishing of the factor

(x~n - x~n')' independent of the value of the integral. To find the

value of the norm of the functions (35) we use (18) of Appendix A. Then we have 1

f

zJt(X/J.nZ ) dz

=t J t+l(X/J.n)

(36)

o

In terms of the notation of Sec. 2-2 the orthonormal Bessel functions -1 fixed. For any value of Jl, these normalized Bessel functions can be used as the expansion functions in (19) with coefficients (15). Often the normalized functions (37) are not used explicitly. Rather the series expansion on the interval (0,1) for an arbitrary function f(z) is written as

ORTHOGONAL FUNCTIONS

f(z)

=

f

33 (38)

AjJ.n J jJ.(x jJ.nz)

n==1 From (15) and (37) it is easy to show that the coefficients AJln are given by 1

AjJ.n

= J2

2(

)

Jl + 1 x Jln

f zf(z)J

J.1. (x J.1.n z )

dz

(39)

0

The form (38) and (39) is known as a Fourier-BesseL series. Problem: (a) Show that the functions {Z J Jl (xMnz) are

orthogonal on the interval (0,1) when the x Mn are the roots of J/J.(x) = O. (b) Find the norm of these ort~l func- • tions and write down explicitly the modified Fourier-Bessel series equivalent to (38) and (39). (c) Discuss the types of function f(z) that will have more rapidly converging expansions in terms of the present series than in terms of (38) and (39). The Fourier-Bessel expansion (38) can be employed to solve the initial-value problem of the vibrating membrane, just as was the Fourier expansion for the vibrating string at the end of Sec. 3-1. Problem: Prove that on the interval (0,1)

z ==

L; n==1

By using numerical tables, find the actual values of the first three nonvanishing coefficients. 3-5

SCHMIDT ORTHOGONALIZATION METHOD

In the preceding sections it has been assumed that the orthogonal functions (a" -

a') ,.

(90)

rather than Kronecker delta normalization, as in (7). Summation~ over a' are replaced by integrals, and so on. Thus, for a vector ~, its expans ion in terms of the bas is cP , is a

~= JO da' cP ,(cp "~) a a

(91)

For a variable a' with mixed discrete and continuous ranges, the expansion relation (91) has an obvious generalization. Before establishing contact with the orthonormal functions and eigenvalue problems of Chapters. 3 and 4 we wish to introduce some very useful notation, due to. Dirac. In dealing with vectors with complex sCc\lar products we have had to be careful about the order o.f f~l,ctors and conlplox c'onJutl:ut:l.ol1 le.fl;., t.hE.) HCltLur product

74

MATHEMATICS FOR QUANTUM MECHANICS

(CPa'CPa) ~ cplCPa and the completeness relation (16)]. To elucidate this distinction Dirac considers all possible vectors ~ in a vector space (with column matrices as representatives), and denotes the typical vector by the symbol I) or 1 ~). Corresponding to the space spanned by these vectors, 1 ~) is a dual space spanned by dual vectors (with row matrices as representatives), which Dirac denotes by (lor (~I. For each vector 1 ~) (Dir~c calls it a "ket" or a "ket vector") there is a dual vector (~I (Dirac calls the dual a "bra" or a "bra vector"), and vice versa. The superposition of these two kinds of related vectors follows the rule

while multiplication by a complex scalar has the correspondence al~)'--

a* (~I

The action of operators on the ket vectors is as we have already se~n in Secs. 5- 5 and 5- 6, with correspondences in the dual space of the form 111 )

= A I~)

A( 1~l)

+ 1~2) )

=AI~l)

+ A 1 ~2)

A(a 1~») = aA 1~ )

}

{

--

«

(7J1

=(~IAt

~ll

+ (~21)A t

= (~lIAt

+ (~21 At

(94)

(~la*)At = a*(~IAt

In Dirac's notation the scalar product becomes (~, 11) = ( ~ 111)

(95)

with the obvious condition (~ 111) = (171 ~)*. As another illustration w~ note that a Hermitean operator is now specified by (96)

Problem: In the scalar product ( ~ 1K 111), show explicitly that the operator can be considered to be acting either to the right on the ket 111) or to the left on the bra ( ~ 1• Another piece of useful notation is the convention that operators are specified by Greek or Latin letters, while their eigenvalues are, specified by the same symbol with a prime (or several primes) on it. Specific eigenvalues may have the prime replaced by a iUbScri~t

LINEAR VECTOR SPACES

75

index. To illustrate this notation we write the eigenvalue equation (73) in the Dirac notation: K I K')

= K' I K' )

In this ~quation the eigenvalue of the operator K is K' and the corresponding eigenvector is written as a ket vector with the eigenvalue inside. By comparison with the eigenvector symbol '11 a, we see that the prime replaces the ordering index a and the eigenvalue symbol K' inside the ket replaces the '11 symbol which distinguished the eigenvectors of K from some other set (e.g., the basis CPa). The economy and clarity of the Dirac notation should be readily apparent. All the material of the previous sections of this chapter can be transcribed into the Dirac notation in a straightforward manner. Some particularly illuminating examples of Dirac notation are the completeness relation (16), ~

la') (a'i = 1

(98)

a' the expansion (11). of an arbitrary ket vector I ~) in terms of a basis

I a') ,

I~)=~ la')(a'I~)

(99)

a' and tile projection operator P €' ~efined by (47), (100) In equations (98) and (100) the operator nature of the entities is clearly displayed by the bras and kets in reverse order, waiting to make scalar products with some vector. 5-11

STATE VECTORS AND WAVE FUNCTIONS

So far we have dealt with abstract vectors and operators or. their matrix representatives. The reader with- some familiarity with quantum mechanics via the Schrodinger equation may wonder why there has been no mention of a wave junction, since the avowed purpose of these notes is to introduce the mathematics of quantum mechanics. The reason is that in a finite-dimensional space the reprepresentatives of state vectors are not functions (of continuous variables), but rather column vectors (with discrete elements), These column vectorH are nllscent w~\.ve fUllcti.ons.

76

MATHEMATICS FOR QUANTUM MECHANICS

To show how a wave function arises ·in Hilbert space we first consider a Hermitean operator q which has a continuum of real eigenvalues q' on some interval. Then the ket eigenvectors 1 q') satisfy the operator equation, q I q')

= q' I q')

(101)

with orthonormality, ( q" I q')

= 6(q' - q")

(102)

Now consider an arbitrary ket vector I ~). It can be expanded in terms of the q bas is (eigenkets of the operator q):

10 =

f dq' 1q') (q' 10

(103)

The scalar product (q' I ~) is a complex number depending on the continuous variable q' and on the original ket 1 ~). It can then be thought of as a complex function of q', which is the representative of the ket vector 1 ~) in the basis 1 q'). We call such a representative a wave junction and write it as '11 ~(q')

= (q'

I~)

(104)

The connection with the functions of Chapters 3 and 4 can now be established. As an example, we show the connection between the scalar product (~ 111 ) and the inner product of Sec. 3-2. We merely write out the scalar product in terms of the expansion (103):

(~I17) =

jdq' jdq"

(~Iq')

(q'lq") (q"I17)

Using the definition of the wave function (104) and the orthonormality (102), we find ( ~ 117)

=

f d.q' wt(q')W17(q')

(105)

This is exactly the inner product of Sec. 3-2.

Problem: Show that in the q basis the operator q is represented by the diagonal "matrix" ~

(q"lqlq') = q'6(q'-q") and that any function F(q) has the representation (q" 1 F(q) 1 q') = F(q')6(q' - q")

/

LINEAR VECTOR SPACES

77

To show the appearance of the wave function in an operator equation and the connection with the linear operators on functions of Sec. 4-2 we consider the' 'equation of motion" (68): i :s

I ~,s) = K I ~,s )

(106)

The set of eigenkets 1 q') is assumed not to depend on the parameter s. Then by taking scalar products of both sides of (106) with the bra vector (q' I, we obtain i :s (q'

I ~,s) = (q' I KI ~,s)

Now we use the expansion theorem (98) or (99) to "insert a complete set of states" to the right of the operator K. This gives i :s (q'

I ~,s) =

f

dq" (q' I K I q") (q"

I ~,s )

(107)

The matrix element (q' 1 K 1 q") is a function of two continuous variables and can be written (q'l K 1 q")

= k(q',q")

(108)

The scalar product (q' I ~,s) is a wave function (104) depending upon two continuous variables (q and s). Thus (107) can be written i :s lit ~(q' ,s)

=

f

dq" k(q' ,q ")lIt ~ (q" ,s) \

(109)

This form of the operator equati~n is entirely analogous to the representation of a linear differentia~r integral operator in Sec. 4-2 _ [see Eq. (7) and b e l o w ] . ' Problem 1: Express the matrix element ( ~ 1 K 111) in terms of the wave-function r.epresentatives in the q basis, in a manner analogous to (105). If the operator K is diagonal (has a diagonal representative) in the q basis, what is the form of (~ 1K 111) in terms of wave functions? Problem 2: The denumerable Fourier basis 1 n') has scalar products with the q basis given by

(q'ln') By oxpl.l.C'1t

VI sill (-n':l()

c~alrulat 1011 liM InK

I)! I"ac' llotJltloll Hhow how

78

MATHEMATICS FOR QUANTUM MECHANICS the wave function 'J1 ~(q') of a vector I ~) in the q basis can be expressed as a series expansion with respect to the Fourier basis.

To complete our discussion of wave functions we shall show that wave functions can be interpreted as elements of a unitary transformation matrix, albeit a matrix with at least one continuou's index. We shall consider the eigenvalue problem (97) in terms of representatives with respect to the continuous q basis (101). By steps closely similar to those from (106) to (109), we find the q representative of (97) to be

f

dq" k(q' ,q")w K,(q") = K'w K,(q')

(110)

where k(q' ,q") is given by (108) and the wave function 'J1 K ,(q') = (q' I K') is the q representative of the eigenvector I K') . In Sec. 5-9 the eigenvalue problem K'J1 a = Aa 'J1 a took the form of determining the eigenvalues and the unitary matrix U from the set of simultaneous linear equations (83): (83) It was found that the columns of U (i.e., ua {3' a = 1,2, ... ,n; (j fixed) were the representatives of the eigenvectors 'J1(j in the original cp basis. Comparison of (110) and (83), with allowance for the continuum integral replacing the discrete sum, shows that there is an equivalence, u

a{j

....-. 'J1 K ,(q')

= ( q' I K' )

(111)

Thus the wave function can be -interpreted as the representative of the unitary operator U: . K'-

D=

~' ( ( q' I K' >

)

Solving an eigenvalue problem with a linear differential or integral equation and obtaining the wave functions is completely equivalent to (in fact, it is) solving for the unitary transformation U which diagonalizes the operator.

Appendix A Bessel Cylinder Functions Unless otherwise specified, the Greek indices and the variable z stand for arbitrary complex numbers, while the Roman indices are nonnegative integers, and the variable x is real. The standard, encyclopedic reference to Bessel functions is the treatise by G. N. Watson, "Theory of Bessel Functions," 2nd ed., Cambridge University Press, New York, 1944. A-l

DEFINITIONS

The Bessel differential equation, 2

d2 1 d 11 ( -dz 2 + -Z -dz + 1 - -Z2

)

Z (z) = 0 II

arises in the separation of the Laplacian operator in cylindrical and spherical coordinates. The functions ZII(Z), which satisfy Bessel's equation, are called cylinder function~f order II. Power-series solutions around the rigin are

_(z)

II

J ,)z)

-"2

00

j~O

( _

1) j

j! r(j +

1/

,I

+ 1)

(z) "2

2j

The function r(z) is the ganuna function (MO, p. 1). For II ~ m, J II and J _II are linearly indepondont solutions of (1). But for II an integer, it crtn be shownf'roln (2) thut

80

MAT HEM A TIC S FOR QUA N TUM M E C HAN I C S

Because of the distinction between II an integer and not for the solutions J II and J -II' it is convenient to use J lI(z) as one independent solution (called a Bessel function, or Bessel function of the first kind) and to define the other independent solution to be the Neumann function (Bessel function of the second kind) N lI(z), given by (4)

For

not an integer, Nil is clearly linearly independent of J II. When m, Nm can be shown to be still linearly independent of J m . L'Hospital's rule can be used with (4) to define Nm(Z) as II

II -

1 [aJ II m aJ- II N (z) = lim - - (- 1) - m lI-m 'IT . all all

]

From the series (2) it is evident that Nm(z) will involve In z, as expected for a differential equation whose indicial equation has roots differing by an integer. The general form is given by MO, p. 17. The function N_m (z) satisfies (3). It is sometimes convenient to introduce another pair of cylinder functions, called Hankel functions, H~)(z) and H~)(z), defined by H~l >(z)

= J lI(z) + iN lI(z) (6)

The Hankel functions form a linearly independent pair of solutions to the Bessel differential equation. A-2

RECURRENCE FORMULAS

· (2) T he f unctIons J II' N II' H(l) I I ' H II are all normalize d so that they satisfy the recursion formulas, Z

11-

1(z) + Z

11+

1(z):=

211

z

Z lI(z) v

(7) Z

11-

1(z) - Z

11+

dZ lI(z) 1(z) := 2 - d z

From these can be derived others, such as

81

BESSEL FUNCTIONS

(8)

A-3

ASYMPTOTIC FORMS

The small and large argument limits of Bessel functions are useful. For simplicity we take the order II to be real and nonnegative.

I z I « (larger of 1 and

No(z)

-~

II)

[In(i) + 0.5772 ... J (9)

N 2 (z) -

-

(2)11 -1TZ r(lI)

I z I » (larger of 1 and

II

~

cos(z _

~1r _~)

N)z) -

~

Sin(z -

11 1r -

II

0

II)

JII(z) -

HO ,21(Z) _

=1=

('f i) II + 1/2

V2

2

~)

(10)

e± iz

1TZ

The transition from the small-argument limiting form to the Iargeargument limit occurs in the region I z I II. '"V

A-4

ROOTS OF BESSEL FUNCTIONS

It is evident from (10) that both J II and Nil have an infinite number of roots. Tables A-1 and A-2 list a few of the smallest roots for II = 0,1,2. More extensive tables of roots can be found in Watson's "Bessel Functions," pp. 748ff. Sometimes the maxima and minima of J lI(z) are wanted, rather than 'the

.J' (x) nl

zor()~.

o.

Table A-3 liRts u fow of tho rootH of tho cquatl,oI1.s,

MATHEMATICS

82

FOR QUANTUM MECHANICS

Table A-1 Roots of J m (x mn)

=0

n m

1

2

3

0 1 2

2.405 3.832 5.136

5.520 7.016 8.417

8.654 10.173 11.620

Table A-2 Roots of Nm (Ymn) = 0 n m

1 0.8936 2.197 3.384

0 1 2

2

3

3.958 5.430 6.794

7.086 8.596 10.023

Table A-3 Roots of J~(x~n)

=0

n m

1

2

3

0 1 2

0 1.841 3.054

3.832 5.331 6.706

7.016 8.536 9.970

Very complete numerical tables of Bessel functions exist. Jahnke and Emde and Watson have tabulations extensive enough for the average student. Morse and Feshbach, p. 1565 and Appendix, also give short tables.

A-5

MODIFIED BESSEL FUNCTIONS

Solutions of the differential equation d2

[-

dz 2

2 1 d 11 + - - - 1- -

Z

dz

Z2

]

n II (z)

= 0

(11)

BESSEL FUNCTIONS

83

are called modified Bessel functions. Comparison with (1) shows that they are related to Bessel functions of argument iz, instead of z. It is customary to define the two, linearly independent, modified Bessel functions (of the first and second kind), I1,,(Z) and K1/z), as IlI(z)

= i -ll J lI(iz) (12)

It can be shown that K_ lI (z) = KlI(z). If 1I and z are real, then IlI(z) and KlI(z) are also real; they are not oscillatory, but monotonic, functions of z. The recursion formulas satisfied by III and K lI are I

- I

11-1

K

1I-1

1I+1

-K

= 211 I

1I+1

z

1I

211 =--K z 1I

I 1I- 1 + I 11+ 1

= 21'1I

K 1I- 1 + K 1I+ 1

= -2K'II

The limiting forms for small and large argument (with nonnegative) are

Iz I «

II

real and

(larger of 1 and II)

- [In(i) + 0.5772 ...J

(14)

1I"* 0

x

»

(larger of 1 and II)

k

II/(X) K (x) 1I

A-a

-+

Vr;h

1

eX e- x

INDEFINITE INTEGRALS

From

tlH)

~of any 't,wo

cUffortHl'ttnl (]qun'lion (1) 'lhE~ l'ollowinp; Indefinite lntegrtll (~yllfld(H' fUllc'llollH (.l.'I,N II ,H:),a») can hE.' obtutncd:

O'll'fUJtfl'It',V

84

MATHEMATICS

FOR QUANTUM MECHANICS

The prime denotes differentiation with respect to the argument. For the modified Bessel functions, the corresponding integral is

(17)

If the arguments and orders are the same, the following indefinite integral holds for ordinary cylinder functions:

JzZ II(Z)~II(Z)

dz

= ~2

[Z~~~ + (1 - ~:) ZII~II ]

(18)

The corresponding integral for the modified Bessel functions is

A-7

SPHERICAL BESSEL FUNCTIONS

In the separation of the wave ~quation, (v 2 + k 2)'IJ coordinates, the radial equation takes the form

~ [~ dr + r 2

~ k2 dr +

-

f

(~r +2 1) ]

f ( ) f r

=0

= 0,

in spherical

(20)

The solution is

Because of the importance of these functions it is c-ustomary to define spherical Bessel and Hankel functions, denoted by j~ (z); n~ (z), h~1,21 (z), as follows: (-~

85

BESSEL FUNCTIONS

ng{z)

=

(;zt

2

Nf+t(Z)

h~I,2) (z) = j f (z) ± in (z) f

For z 'real, h~2) (x) is the complex conjugate of h;l) (x). From the series expansion (2) it is possible to show that

~) f dz

(1:z

j (z) = (_ z)! f

:Z)

(_z)f(~

nf{z) = -

(sin z ) z

For the first few values of

~

f

Z)

(CO:

the explicit forms are

z -sin --

no(z) = _ cos z z

z

sin z

cos z

=~-Z2

(1)(

hI

= 2:

f

z

)

iz (

e =- Z

1

+z

C

~)

sin

3

h(z)

=

n2(z)

= -

3 -

1 0

)

Z -

3

Cz~s

(24) Z

z -(3 1) cos z - -3sin -

Z3

- -

Z

Z2

3 3)

e iz ( 1 1 +- - z Z Z2 0

(1)

h 2 ·(z) =

0

1-

From the general asymptotic forms, (9) and (10), the asymptotic forms of the spherical Bessel functions can be found to be

I z·1

«' (larger of I.

and 1)

zq jg(z) --. (2q ·1· 1}11

n (z) _ _ (2~ - 1) II ~

Z

~

·1,·

1

86

MATHEMATICS

FOR QUANTUM MECHANICS

where (2f + 1)! 1 = (2~ + 1) (2f - 1) (2! - 3) ( • • • ) x 5 x 3 x 1.

I z I » I. jll (z)

---

~

n! (z) --- -

sin ( z -

~

g2'") (26)

cos (z - !2'" )

Wronskians of various pairs of spherical Bessel functions are

Some recurrence formulas for spherical Bessel functions are

d r.~+1 ] dzLz ~f(z)

=Z

f+1

d~ [z- g ~! (z)] = -

~f_1(z)

z- g

~! + 1(z)

where ~~ (z) is any linear combination of j f' n , h~l) , h~2) . f Since the spherical Bessel functions are intimately connected to ordinary Bessel functions, various definite and indefinite integrals can be generated from equations (16) through 1(19) above. A useful example is the indefinite integral

JZ2~!

(z) z! (z) dz

= ~3

[~f Zg - ~ ( ~! -1 Zh

~

1 + h 1Z! -1) ]

where ~! and Z! are any linear combinations of the s_pherical Bessel functions. For! = 0, the functions ~-l and Z-l must be converted according to the rule [obtained from the third recurrence relation in (28)], / /

BESSEL FUNCTIONS

87 h(l)

-1

=

ih0

(30)

The normalization of the regular spherical Bessel functions jf(kx) on the infinite interval (0 ~ x < 00) is delta function normalization, 00

J

x j f (lac) j f (k' x: dx = 2~2 6(k -k') 2

o

A brief table of the rods of jf (x)

= 0 is given in Table A-4. Numeri-

cal tables of j f (x) are available in Mathematical Tables Proj ect, Natl. Bur. Standards, "Tables of Spherical Bessel Functions," 2 vols., Columbia University Press, New York, 1947. Table A-4 Roots x~ m of jf (x) = 0 m

.

f

1

2

3

4

0 1 2 3

3.142 4.493 5.763 6.988

6.283 7.725 9.095 10.417

9.425 10.904 12.323 13.698

12.566 14.066 15.515 16.924

Appendix B Legendre Functions and Spherical Harmonics The general notation outlined at the beginning of Appendix A will apply here. More detailed information will be found in MO and in the mathematical references cited in Chapter 1.

B-1

DEFINITIONS

The Legendre differential equation

occurs in the separation of the Laplacian operator in spherical coordinates. In that context, the variable z = cos e. In the general case, J.l and ZJ are arbitrary complex numbers. But in the most common physical applications, J.l and ZJ are both integers. The solutions of (1) are called Legendre functions, or more precisely, Legendre functions if J.l = 0 and associated Legendre functions if J.l "* o. Since the Legendre differential equation is second order, there are two linearly independent solutions, denoted by pJ.l(z) and QJ.l(z), ZJ

ZJ

and called Legendre functions of the first and second kind. If J.l = 0, the solutions are denoted by P ZJ(z) and QZJ(z).

B-2

LEGENDRE POLYNOMIALS

If J.l = 0 and ZJ =~, where t = 0,1,2, ... , one solution of Legendre's equation is finite, single-valued, and continuous .on the interval -1 ~ x ~ 1. This solution, denoted by P~ (x), and normalized to the value unity at x = 1, is called a Legendre polynomial of order~-:~A

LEGENDRE

FUNCTIONS

89

power-series solution of (1) shows that, for the first few values of P, these Legendre polynomials are

A general formula for p~(x) is Rodrigues' formula: P (x) II

=

+ 2 £!

f d

dx~

(x2

-

l)f

Some special values are

(4)

where (2~ - 1)! ! = (2£ - 1) (2P - 3) (2~ - 5) . . . (5) (3) (1). B- 3 RECURRENCE FORMULAS FOR P I. (x)

dP

" dP

~+1 _

dx

--!:l dx

(2~ + 1)P = 0 ~

(6)

dP~+ 1 dP~ -~ - x dx - (g + 1)Pg

=0 (R)

90

MATHEMATICS

B-4

FOR QUANTUM MECHANICS

ASYMPTOTIC FORMS FOR p!(cos 6)

In the limiting case of large order (~ » 1) it is sometimes convenient to make use of the asymptotic formulas (0 ~

e«

1)

and

(e» i) B- 5

(10)

LEGENDRE FUNCTIONS Q! (x) OF THE SECOND KIND

The other linearly independent solution Q~ (x) on the interval (-1,1) for ZJ = Q, J.l = 0 involves logarithms and diverges at x = ± 1. The general form of the second solution can be found by the Wronskian method (MO, p. 160). The standard definition of Q (x) is

t

Q

Qf(x)

=

~ Pf (x)

In( ~ ~ ~) - L

~

Pm -1 (x)Pf _ m (x)

(11)

m=1 For ~ = 0 the summation term is defined to be zero. The first few Q~ (x) are Qo(x)

1In (11 -+ x) x

="2

( ="2XIn (1+X) 1_ x -

Ql x)

B-6

(12)

1

ASSOCIATED LEGENDRE POLYNOMIALS

P~(x)

The solutions of (1) when ZJ = t, where t = 0,1,2,···, and J.l = m (m an integer) are of considerable importance. If the solution is to be finite, single-valued, and continuous on the int~rval-1 ~ x ~ 1, it is necessary that m be confined to the range - t ~ m ~ t. Thus, , for fixed Q, there are (2~ + 1) well-behaved functions pr(x~. For positive m, the associated Legendre function is defined by / -----------

!

91

LEGENDRE FUNCTIONS

[The associated function Q~(x) is defined from Q (x) in exactly the lI same way.] For negative m, the solution p;m(x) is related to p~(x) according to P ~-

m( ) x

= (_1)m

(t - m)! p m ( )

(t + m) 1

(14)

t x

For all values of m (- t ~ m ~ t) the generalized Rodrigues' formula obtained by substituting (3) in (13) holds (15)

For the first few values of t and m functions pr(x) are

> 0,

the associated Legendre

(16)

The phase choice of p~(x) in (13) through (16) is that of MO and that of E. U. Condon and G. H. Shortley, "Theory of Atomic Spectra," Cambridge University Press, New York, 1953. Recurrence formulas for associated Legendre polynomials are given by MO on p. 54. These recurrence relations connect functions of the same ~, but neighboring m values, or functions of the same m, but neighboring ~ values.

B-7

INDEFINITE INTEGRA~S AND ORTHOGONALITY; NORMALIZATION INTEGRALS

From the differential equation (1) the following indefinite integral of two solutior18 ZIJ.(z) and EIJ.,'(z) cnn be established:" II

II

92

MATHEMATICS

FOR QUANTUM MECHANICS

(17)

Equation (17) can be used to establish the orthogonality and norms of the P~(x) on the interval (-1,1). Since p~ is finite and has a fi-

nite slope everywhere on the interval (-1,1), including the end points, the right-hand side of (17) vanishes at x = ± 1. Hence, with IJ.' = IJ. =m and II = t, II' = t', (17) gives the orthogonality of the pf's for the same m, but different t's: 1

[f(f + 1) - f'(£' + 1)]

f

P~(x)P~(x)

dx

=0

(18)

-1

The norm of the function pf(x) on the interval (-1,1) can be established by means of (15). The result can be expressed as an orthogonality- normalization integral,

(19)

For m = 0, (19) gives the norm noted in Sec. 3-6, Eq. (51). B-8

EXPANSION IN SERIES OF LEGENDRE FUNCTIONS; COMPLETENESS RELATION

The set of functions

P~(x), wi~h m fixed and

f

= m, m + 1,

m + 2, .•• ,

form a complete orthogonal set on the interval (-1,1). An arbitrary function f(x) can be expanded, as in Sec. 3-2, in the series

f(x)

=

~ a~p~(x)

(20)

~=m

with expansion coefficients [in view of (19)] m

a~

= 2~ + 1 (~ - m)! 2

(~ + m)!

f

1

p m ( ) f( ) dx t x X

-1

The completeness relation for the pr(x) is

(21)

LEGENDRE

FUNCTIONS

93

00

2t + 1 (t - m) 1 pm( )pm( ') 2 (t + m) 1 t X t x

'\'

L

= ~( _ ') u

X

x

~=m

B-9

SPHERICAL HARMONICS Y!m (8,cp)

In spherical coordinates it is useful to have a set of orthonormal functions in the angular variablest (e,cp), where ~ 8 ~ 'IT, ~ cp ~ 2."., For short, we sayan orthonormal set on the unit sphere. imcp The Fourier series functions, e (m = 0,±1,±2, ...), form a complete orthogonal set in the '{ariable cp. The associated Legendre polynomials Pf(cos 8), m fixed, ~ = m, m + 1, ... , form a complete or-

°

thogonal set in the variable cos

Hence the product,

P~(cos e)e

e on the range

imcp

(- 1 ~ cos

°

e~

1).

, will form a complete orthogonal

set on the unit sphere. The orthonormal functions, denoted by Y (e,cp) and called spherical harmonics, are tm

_V2t~ + 1 (t - m)! (t + m)t

Y~m(e,cp) -

m imcp P t (cos 8)e

The ranges of the indices t and mare t = 0,1,2, ... , and m = -~, - (t - 1), ... , -1,0,1, ..., (~ - 1),~. The spherical harmonic for negative m is related to the complex conjugate of that for positive m, according to (24)

The orthonormality conditiDn is

2'IT

f

o

'IT

dcp

J

sin e de Y;'m,(e,cp)Y£m (e,cp)

= l)H' l)mm'

(25)

0

The completeness relation is 00

L ~=o

t ~ Y;m(e',cp')ytm(e,cp) m=-t

= 6(cp -

cp')o(cos

e ~ cos e')

tActually the variables are (cos 8, cp) since we are concerned with spherical coordinates where the volume element is d 3x = r i dr dO, with dO ~ d(cos 8) dcp ..

94

MATHEMATICS FOR QUANTUM MECHANICS

For the lowest values of t and m ~ 0 explicit expressions for the spherical harmonics are given below. For negative m values, (24) can be used. 1

t = 0: Yoo = .f4i £

= 1:

Y10

=

~

lr3 icp V&r sin e e

Y11 = -

£

= 2:

Yzo

= 3:

cos

e - ~)

2

, Ii5 .

v87T sIn e cos e e icp 1 1m . e 2icp ="4 V~ sIn e 2

22

t

a

=~

Y2 1 = -

Y

e

cos

Y 30

= lr'7(5 V j-:;411' 2

Y 31

= - "41 1/21 V~

Y 32

="4

1

llI05

cos 3

(27)

. e (5 cos 2 e - 1) e icp

SIn

.

V"21t

e - -23 cos e)

SIn

2

e cos e e 2icp

1 1~ • 3 3icp Y33 = - - v:r:;- SIn e e 4

~

= 4:

41T

-41T ~

Y40

=

Y41

=-

Y42

=

Y43

= -"4

~

4 15 2 3) 35 ( -8 cos e- - 4 cos e +8

~~

sin

y.;,;

2

sin

3,/35

Y 44

V~

e(7

cos

s

e-

e (7 cos e -

. 3 SIn

2

3 cos

1) e

e)e icp

2icp

e cos e e 3icp

3,135. 4 e 4icp SIn e

="8 V87T

As already mentioned, the choices of phase for Yl m are those of Condon and Shortley. The Condon-Shortley phase conventio~ is-the

LEGENDRE

FUNCTIONS

95

most common one, but others are in use. We list some of the more common references on quantum mechanics and their phase convention in Table B-1. The factor listed in each case is that by which the present definition must be multiplied in order to get the form used in the corresponding reference. Table B-1 Phase factor (relative to C-S)

Reference H. A. Bethe and E. E. Salpeter, "Quantum Mechanics of One- and Two-Electron Atoms," Academic, New York, 1957

(none given) 1

L. D. Landau and E. M. Lifshitz, "Quantum Mechanics," Addison-Wesley, Reading, Mass., 1958

1

E. Merzbacher, "Quantum Mechanics," Wiley, New York, 1961

1

A. Messiah, "Quantum Mechanics," Vols. 1 and 2, Interscience, New York, 1961-1962

1

J. L. Powell and B. Crasemann, "Quantum Mechanics," Addison-Wesley, Reading, Mass. 1961

( _ 1) ! (m + I m

B-10

D. Bohm, "Quantum Theory," Prentice-Hall, Englewood Cliffs, N.J.. , 1951

I)

L. I. Schiff, "Quantum Mechanics," 2nd ed., McGraw-Hill, New York, 1955

ADDITION THEOREM FOR SPHERICAL HARMONICS

If y, is the angle between two vectors whose angular positions are specified by (e,cp) and (e',cp'), as shown in Figure B-1, then the Legendre polynomial P t (cos y), with cos y = cos e cos e' + sin e sin e' x cos(cp - cp'), may be written as a bilinear expansion in spherical · harmonics of order t : Pf(cos y)

=u

4

t : 1

E

Y;m(e',cp')yfm(e,cp)

(28)

m=-t This is called the addition theore'n1 for spherical harmonics. Equation (27) can be derived tn it, nonri,~orollH way by noting that the

96

MATHEMATICS

FOR QUANTUM MECHANICS

z

Iy I

I I

~-----_----:.-

I -1-................

x

"

I

'J

y

I -"'-J

Figure B-1

product of delta functions on the right side of the completeness relation (26) can be written 1

o(cp - cp')o(cos 8 - cos 8') = 21T o(cos y)

Then the completeness relation (22) for Pp's can be used to represent o(cos y) in the right side of (26). Comparison of the individual t terms on each side yields (28). If y - 0, equation (28) reduces to a sum rule for spherical harmonics, ~

~

L.J

m=-t

IYtm (8 ,cp )1 2 =2t+l 41T

This can be interpreted to mean that the average over m of the absolute square of Ytm(8,cp) is spherically symmetric and equal to the absolute square of Yoo. B-11

USEFUL RECURRENCE RELATION FOR SPHERICAL HARMONICS

A general expansion of the product of two spherical harmonics in a series of single spherical harmonics is

x

(~~'OOI ~~'LO)(tQ'mm'IP~'LM) YLM(8,cp)!

(30)

LEGENDRE FUNCTIONS

97

The coefficients (~~ 'mm' I Q~ 'LM) are called Clebsch-Gordan or Wigner or vector addition coefficients, and are defined in E. U. Condon and G. H. Shortley, "Theory of Atomic Spectra," Cambridge University Press, New York, 1953, pp. 73-78, or A. R. Edmonds, "Angular Momentum in Quantum Mechanics," Princeton University Press, Princeton, N.J., 1957, p. 52. Two useful special cases, explicitly exhibited, are 1

I

COS8Ytm=v'2l+1

1/(~

+ V' sin 8

e

[1/~2-m2

+ 1)2 - m 2 2t + 3

±~CPY

±

J

YQ + 1,m

1 tm -± v'2b +

_ VC~

Y~-1,m

V2t-1

i

[1/(t V

=F

m)(t =F m - 1) 2P -1 Yt - 1,m±1

m + 2W. ± m + 1) y ] 2t + 3 t + 1, m ± 1