Mathematics for quantum mechanics; an introductory survey of operators, eigenvalues, and linear vector spaces

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 www.pdfgrip.com 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 www.pdfgrip.com 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, www.pdfgrip.com 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 www.pdfgrip.com 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 www.pdfgrip.com www.pdfgrip.com Contents Editors' Foreword v Preface vii Introductory Remarks References 2 Eigenvalue Problems in Classical Physics 2-1 Vibrating string 2-2 Vibrating circular membrane 2- Small oscillations of a mechanical system 10 2- 15 Rotation of axes and orthogonal transformations 2- Euler's theorem and principal-axes transformations as eigenvalue problems Orthogonal Functions and Expansions 18 )2 3-1 Fourier series 22 3-2 Expansion in orthonormal functions 25 3- Dirac delta function and closure relation 28 3- Bessel functions as an orthonormal set on the interval (0,1) 31 3- Schmidt orthogonalization method 33 3-6 Legendre polynomials 35 www.pdfgrip.com CONTENTS x 3-7 Other orthogonal polynomials 36 3- Fourier integrals 37 Sturm-Liouville Theory and Linear Operators on Functions 41 4-1 Sturm-Liouville eigenvalue problem 41 4-2 Linear operators on functions 43 4- Eigenvalue problem for a linear Hermitean operator 45 Further properties of operators 46 4-4 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- 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 www.pdfgrip.com Mathematics for Quantum Mechanics www.pdfgrip.com 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 I 1I- + I 11+ 1I 211 = K z 1I = 21'1I K 1I- + K 1I+ = -2K'II The limiting forms for small and large argument (with nonnegative) are Iz I « II real and (larger of and II) - [In(i) + 0.5772 J (14) 1I"* x » (larger of and II) k II/(X) K (x) 1I A-a -+ Vr;h 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 www.pdfgrip.com 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 + k 2)'IJ coordinates, the radial equation takes the form ~ [~ dr + r ~ dr + k2 - f (~ + 1) ] f ( ) = r2 f r = 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: (-~ www.pdfgrip.com 85 BESSEL FUNCTIONS ng{z) = (;zt 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 +z C ~) sin h(z) = n2(z) = - - ) Z - Cz~s (24) Z z -(3 1) cos z - -3sin - Z3 - - Z Z2 3) e iz ( 1 +- - z Z Z2 (1) h ·(z) = 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,· www.pdfgrip.com 86 MATHEMATICS FOR QUANTUM MECHANICS where (2f + 1)! = (2~ + 1) (2f - 1) (2! - 3) ( • • • ) x x x 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 ~ + 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)], / / www.pdfgrip.com 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') o A brief table of the rods of jf (x) = 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," vols., Columbia University Press, New York, 1947 Table A-4 Roots x~ m of jf (x) = m f 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 www.pdfgrip.com 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 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 = 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 = and ZJ =~, where t = 0,1,2, , one solution of Legendre's equation is finite, single-valued, and continuous on the interval -1 ~ x ~ This solution, denoted by P~ (x), and normalized to the value unity at x = 1, is called a Legendre polynomial of order~-:~A www.pdfgrip.com 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 = + £! f d dx~ (x2 - l)f Some special values are (4) where (2~ - 1)! ! = (2£ - 1) (2P - 3) (2~ - 5) (5) (3) (1) B- RECURRENCE FORMULAS FOR P I (x) dP " dP ~+1 _ dx !:l dx (2~ + 1)P = ~ (6) dP~+ dP~ -~ - x dx - (g + 1)Pg =0 (R) www.pdfgrip.com 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- (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 = involves logarithms and diverges at x = ± 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 ~ = 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) 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 / - ! www.pdfgrip.com 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) (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 www.pdfgrip.com 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 = ± 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: [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~ + (~ - m)! (~ + m)! f p m ( ) f( ) dx t x X -1 The completeness relation for the pr(x) is www.pdfgrip.com (21) LEGENDRE FUNCTIONS 93 00 2t + (t - m) pm( )pm( ') (t + m) 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 ~ ~ '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 (- ~ 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~ + (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) 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 www.pdfgrip.com 94 MATHEMATICS FOR QUANTUM MECHANICS For the lowest values of t and m ~ explicit expressions for the spherical harmonics are given below For negative m values, (24) can be used t = 0: Yoo = f4i £ = 1: Y10 = ~ lr3 icp V&r sin e e Y11 = - £ = 2: Yzo = 3: cos e - ~) , Ii5 v87T sIn e cos e e icp 1m e 2icp ="4 V~ sIn e 22 t a =~ Y2 = - Y e cos Y 30 = lr'7(5 V j-:;411' Y 31 = - "41 1/21 V~ Y 32 ="4 llI05 cos (27) e (5 cos e - 1) e icp SIn V"21t e - -23 cos e) SIn e cos e e 2icp 1~ • 3icp Y33 = - - v:r:;- SIn e e ~ = 4: 41T -41T ~ Y40 = Y41 =- Y42 = Y43 = -"4 ~ 15 3) 35 ( -8 cos e- - cos e +8 ~~ sin y.;,; sin 3,/35 Y 44 V~ e(7 cos s e- e (7 cos e - SIn cos 1) e e)e icp 2icp e cos e e 3icp 3,135 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 www.pdfgrip.com 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) L D Landau and E M Lifshitz, "Quantum Mechanics," Addison-Wesley, Reading, Mass., 1958 E Merzbacher, "Quantum Mechanics," Wiley, New York, 1961 A Messiah, "Quantum Mechanics," Vols and 2, Interscience, New York, 1961-1962 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 t : 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 www.pdfgrip.com 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 o(cp - cp')o(cos - 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 =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)! www.pdfgrip.com (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 I COS8Ytm=v'2l+1 1/(~ + V' sin e [1/~2-m2 + 1)2 - m 2t + ±~CPY ± J YQ + 1,m 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 + t + 1, m ± www.pdfgrip.com ...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 www.pdfgrip.com 1962 MATHEMATICS. .. properties of operators 46 4-4 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... Rotation of axes and orthogonal transformations 2- Euler's theorem and principal-axes transformations as eigenvalue problems Orthogonal Functions and Expansions 18 )2 3-1 Fourier series 22 3-2 Expansion