1 INTRODUCTION TO QUANTUM MECIIANICS ー ″ ん ノ 盟 盟 ん c 住 0 れ s C ん e 襯 な ケ BY LINUS PAULING, PH.D. , Sc. D. Pro. / e880r 0 / C ん e 襯 む t , C 記 ゾ 0 ァ れ 1 れ e 0 / Tec ん れ 叩 4 AND . BRIGHT WILSON, JR. , PH. D. 880C を 0 ~ ア 0 / 圄 80r 可 C ん e レ リ , 丑 rvard U れ を ~ e ア 8 4 INTERNATIONAL STUDENT EDITION MCGRAW-HILL BOOK COMPANY, INC. NEW YORK ST. LOUIS SAN FRANCISCO LONDON MEXICO PANAMA SYDNEY TORONTO KöGAKUSHA COMPANY, LTD. TOKYO

INTRODUCTION TO QUANTUM MECHANICS INTERNATIONAL STUDENT EDITION Exc ん 5 ど g ん 5 わ ) KÖga た レ 5 ん 0 CO. , . , for 襯 の ct リ の ど x 々 0 れ ″ om 阨 カ ル T ん な し 00 ん cannot と ど - ど x 々 0 滝 ど d from ど ( 0 社 ” 膀 ) こ 0 zv ん 耘 ん れ な co ” g れ ど d と ) Köga い ん 4 CO. , . , or と ) G ァ aw - 〃 澀 お 00 々 Com カ 4 〃 ル ル 佖 れ 工 0 れ ん 0 こ e を ss 乞 0 れ 0 工 佖 s こ ん e eq 工 , 佖 0 こ be ア e od ce ″ ? ・ ル ん こ s 們 お e ァ d. T ん な 600 た , 0 MCGRAW-HILL BOOK COMPANY, INC. COPYRIGHT, 1935 , BY THE VIII 1 れ C. , or any 0 / れ 5 su と 5 ~ d ー a ? ゾ ど 5. こ ん e ん e s. TOSHO PRINTING CO. , LTD. , TOKYO, JAPAN

PREFACE ln writing this b00k we have attempted to produce a textbook Of practical quantum mechanics for the chemist, the experi- mental physicist, and the beginning student of theoretical physics. The b00k is not intended to provide a critical discus- Sion Of quantum mechanics, nor even t0 present thorough survey 0f the subject. We hope that it does give a lucid and easily understandable introduction to a limited portion of quantum-mechanical theory; namely, that portion usually suggested by the name “ wave mechanics,' consisting 0f the discussion 0f the Schrödinger wave equation and the problems which can be treated by means of it. The effort has been made t0 provide for the reader a means of equipping himself with a practical grasp 0f this subject, so that he can apply quantum mechanics t0 most of the chemical and physical problems which may confront him. The book is particularly designed for study by men without extensive previous experience With advanced mathematics, such as chemists interested ⅲ the subject because of its chemical applications. We have assumed 0 Ⅱ the part of the reader, ⅲ addition tO elementary mathematics through the calculus, only some knowledge Of complex quantities, ordinary differential equations, and the technique 0f partial differentiation. lt may be desirable that a book written for the reader not adept at mathematics be richer in equations than one intended for the mathematician ; for the mathematician can follow a sketchy derivation with ease, whereas if the less adept reader is to be led safely through the usually straightforward but sometimes rather complicated derivations of quantum mechanics a firm guiding hand must be kept 0 Ⅱ him Quantum mechanics is essentially mathematical ⅲ character, and an understanding of the subject without a thorough knowledge of the mathematical methods in olved and the results 0f their application cannot be obtained. The student not thoroughly trained ⅲ the theory 0f partial d1 erential equations and orthogonal functions must

INTRODUCTION TO QUANTUM MECHANICS CHAPTER I SURVEY OF CLASSICAL MECHANICS The subj ect Of quantum mechanics constitutes the most recent step ⅲ the very 01d search for the general laws governing the motion 0f matter. For a long time investigators confined their efforts to studying the dynamics of bodies of macroscopic dimen- sions, and while the science Of mechanics remained in that stage it was properly considered a branch 0f physics. Since the development 0f atomic theory there has been a change 0f lt was recognized that the older laws are not correct emphasis. when applied t0 atoms and electrons, without considerable modification. -Moreover, the success which has been obtained ⅲ making the necessary modifications 0f the older laws has also had the result of depriving physics 0f sole claim upon them, since it is now realized that the combining power 0f atoms and, ⅲ fact, all the chemical properties 0f atoms and molecules are explicable ⅲ terms 0f the laws governing the motions 0f the electrons and nuclei composing them. Although it is the modern theory Of quantum mechanics ⅲ which we are primarily interested because 0f its applications t0 chemical problems, it is desirable for us first t0 discuss briefly the background 0f classical mechanics from which it was devel- oped. By SO d0ing we not only f0110W t0 a certain extent the historical development, but we also introduce ⅲ a more familiar form many concepts which are retained ⅲ the later theory. We shall also treat certain problems ⅲ the first few chapters by the methods 0f the 01der theories ⅲ preparation for their later treat- lt is for this reason that the ment by quantum mechanics. student is advised to consider the exercises 0f the first few chapters carefully and t0 retain for later reference the results which are secured. 1

54 ″ 刃 SC 〃 ÖD / Ⅳ G WA 刃 EQ UATION [III -9a e / れ c 0 れ c ん 市 れ 0 the 鹿 忸 % or the probability amplitude function. lt will be noticed that the equation is somewhat similar in form tO the wave occurring ⅱ 1 Other of theoretical physics, as ⅲ the discussion 0f the motion 0f a vibrating string. The student facile in mathematical PhYSiCS may well profit from investigating this similarity and also the analogy between classical mechanics and geometrical 0Ptics 0 Ⅱ the one hand, and wave mechanics and undulatory 0Ptics 0 Ⅱ the other. 1 However, it is not necessary tO d0 this. An extensive previous knowledge 0f partial differential equations and their usual applications in mathematical PhYSiCS is not a necessary prerequisite for the study 0f wave mechanics' and indeed the study Of wave mechanics may provide a satisfactory introduction t0 the subj ect for the more physically minded or chemically minded student. The Schrödinger time equation is closely related t0 the equation Of classical Newtonian mechanics ( 9 ー 2 ) Hamiltonian function 丑 ( , , の . lntroducing the coordinate 工 kinetic energy and the potential energy and hence t0 the which states that the t0tal energy } is equal t0 the sum 0f the and momentum , , this equation becomes 1 ー 十 の = ル . 丑 ( ル , の 2 襯 ( 9 ー 3 ) Ⅱ we now arbitrarily replace , by the differential operator ー , and introduce the function 平@, の 0 Ⅱ and ル by 2 を 2 öx which these operators can operate, this equation becomes ー , ⅵ も の 2 ör ん 2 ö2 872 襯 ö ェ 2 ん öV ( 97 ) 2 which is identical with Equation 9 ー 1. The wave equation is 1 See, for example, Condon and Morse, " Quantum Mechanics"' p ・ 10 , McGraw-HiII B00k Company, lnc. , New York, 1929 ; Ruark and Urey' “ Atoms, M01ecules and Quanta," Chap. XV, McGraw-Hill B00k Compan•Y' lnc. , New York, 1930 ; 刊 . Schrödinger, ユ れ ル d. p ん . 79 , 489 ( 1926 ) ; K. K. Darrow, 側 . 」 一 . p ん い . 6 , 23 ( 1934 ) ; or other treatises 0 Ⅱ wave mechanics' listed at the end 0f this chapter.

Ⅱ ] ア ″ 刃 DECLINE OF ア 刃 0 石 D QUANTUM ア 〃 EO 49 developed by Heisenberg, Born, and Jordan1 by the introduction of matrix methods. ln the meantime Schrödinger had inde- pendently discovered and developed his wave mechanics,2 stimulated by the earlier attribution of a wave character to the electron by de Broglie3 ⅲ 1924. The mathematical identity of matrix mechanics shown Schrödinger4 and by Eckart. 5 The further development of the quantum mechanics was rapid, especially because of the con- tributions of Dirac, who formulated6 a relativistic theory of the electron and contributed to the generalization of the quantum mechanics (Chap. XV). General References 0 Ⅱ the 01d Quantum T eo A. SOMMERFELD: "Atomic Structure and SpectraI Lines, ” 刊 . p. Dutton & Co. , lnc. , New York, 1923. A. . RUARK and Ⅱ . C. UREY.• "Atoms, MoIecules and Quanta, ” McGraw- HiII Book Company, lnc. , New York, 1930. 1 M. BORN and P. JORDAN, 記 . 34 , 858 ( 1925 ) ; M. BO#N, W. HEISENBERG, and P. JORDAN, 防 記 . 35 , 557 ( 1926 ). 2 . SCHRöDINGER, ス れ れ . d. ん い . 79 , 361 , 489 ; 80 , 437 ; 81 , 109 ( 1926 ). 3 L. DE BROGLIE, Thesis, Paris, 1924 ; れ れ . de 〃 ん . ( 10 ) 3 , 22 ( 1925 ). 4 刊 . SCHRöDINGER, ス れ れ . d. P ん い . 79 , 734 ( 1926 ). 5 C. ECKART, P ん い . 側 . 28 , 711 ( 1926 ). 6 P. A. M. DIRAC, Proc. 0 リ . Soc. A 113 , 621 ; 114 , 243 ( 1927 ) ; 117 , 610 ( 1928 ).

CHAPTER III THE SCHRöDINGER WAVE EQUATION WITH THE HARMONIC OSCILLATOR AS AN EXAMPLE ln the preceding chapters we have given a brief discussion 0f the development 0f the theory 0f mechanics before the discovery of the quantum mechanics. NOW we begin the study 0f the quan- tum mechanics itself, starting in this chapter with the Schrödinger wave equation for a system with only one degree 0f freedom, the general principles 0f the theory being illustrated by the special example 0f the harmonic oscillator, which is treated in great detail because of its importance in many physical problems. The theory will then be generalized ⅲ the succeeding chapter t0 systems Of point particles ⅲ three-dimensional space. 9. THE SCHRöDINGER WAVE EQUATION ln the first paragraph of his paperl Quantisierung als Eigen- wertproblem, communicated t0 the れ れ ale れ der P ん リ s on January 27 , 1926 , Erwin Schrödinger stated essentially : ln this communication I wish to show, first for the simplest case of the non-relativistic and unperturbed hydrogen atom, that the usual rules of quantization can be replaced by another postulate, in which there occurs no mention of whole numbers. lnstead, the introduction Of integers anses in the same natural way as, for example, in vibrating string, for which the number Of nodes is integral. The new conception can be generalized, and I believe that it penetrates deeply into the true nature Of the quantum rules. ln this and four other papers, published during the first half of 1926 , Schrödinger communicated his wave equation and applied it t0 a number 0f problems, including the hydrogen atom, the harmonic oscillator, the rigid rotator, the diatomic molecule, and 1 E. SCHRöDINGER, ? 襯 . d. ん s. 79 , 361 ( 1926 ) , and later papers referred to on the preceding page. An EngIish translation of these papers has appeared under the title 刊 . Schrödinger, " Collected papers 0 Ⅱ Wave Mechanics," Blackie and Son, London and Glasgow, 1928. 50

36 T 刃 0 D QUANTUM THEORY [ Ⅱ -7 良 by the discussion Of interference and reinforcement Of waves could be carried through from the corpuscular viewpoint with the 01d quantum theory, and that a similar treatment could be given the scattering of electrons by a crystal, with the introduction 0f the de Broglie wave length for the electron, indicates that the gap between the 01d quantum theory and the new wave mechanics is not so wide as has been customarily assumed. The indefinite- ness 0f the 01d quantum theory arose from its incompleteness— its inability tO deal with any syst ems exc ept multiply-p eriodic ones. Thus in this diffraction problem we are able t0 derive only the simple diffraction equation for an infinite crystal, the interesting questions 0f the width 0f the diffracted beam, the dis- tribution 0f intensity ⅲ different diffraction maxlma, the effect of finite size of the crystal, etc. , being left unanswered. 1 7 THE HYDROGEN ATOM The system composed of a nucleus and one electron, whose treatment underlies any theoretical discussion Of the electronic structure 0f atoms and molecules, was the subject 0f B0hr's first paper 0 Ⅱ the quantum theory. 2 ln this paper he discussed cir- cular orbits 0f the planetary electron about a fixed nucleus. Later3 he took account of the motion of the nucleus as well as the electron about their center of mass and showed that with the consequent introduction of the reduced mass of the two particles a small numerical deviation from a simple relation between the spectral frequencies of hydrogen and ionized helium is satisfactorily explained. Sommerfeld4 then applied his more general rules for quantization, leading t0 quantized elliptical orbits with definite spatial orientations, and showed that the relativistic change m mass of the electron causes a splitting of energy levels correlated with the observed fine structure 0f hydrogenlike spectra. ln this section we shall reproduce the SommerfeId treatment, except for the consideration 0f the rela- tivistic correction. 7a. S01ution of the Equations of Motion. —The system con- sists 0f two particles, the heavy nucleus, with mass 襯 1 and 1 The application of the correspondence prlnciple to this problem was made by P. S. Epstein and P. Ehrenfest, c. Ⅳ 観 . cad. S . 10 , 133 ( 1924 ). 2 N. BOHR, P ん ル 〕 ー 00. 26 , 1 ( 1913 ). 3 N. BOHR, 記 . 27 , 506 ( 1914 ). 4 A. SOMMERFELD, れ . d. ぞ ん . 51 , 1 ( 1916 ).

IV PREFACE learn something 0f these subjects as he studies quantum mechan- ics. ln order that he may d0 so, and that he may follow the discussions given without danger 0f being deflected from the course Of the argument bY inability t0 carry through some minor step, we have avoided the temptation t0 condense the various discussions intO shorter and perhaps more elegant れ S. After introductory chapters on classical mechanics and the 01d quantum theory, we have introduced the Schrödinger wave equation and its physical interpretation 0 Ⅱ a postulatory basiS' and have then given ⅲ great detail the solution 0f the wave equation for important systems (harmonic oscillator' hydrogen atom) and the discussion 0f the wave functions and their proper- ties, omitting none 0f the mathematical steps except the most elementary. A similarly detailed treatment has been given in the discussion 0f perturbation theory, the variation method' the structure 0f simple molecules, and, in general' in every important section 0f the b00k. ln order to limit the size of the b00k, we have omitted from discussion such advanced tOPiCS as transformation general quantum mechanics (aside from brief mention ⅲ the last chapter) , the Dirac theory 0f the quantization of the electromagnetic field, etc. have also omitted several subjects which are ordinarily considered as part 0f elementary quantum mechanics, but which are 0f minor importance t0 the chemist, such as the zeeman effect and magnetic general, the dispersion 0f light and allied phenomena, and rnost 0f the theory 0f aperiodic processes. The authors are severally indebted t0 Professor A. Sommerfeld and Professors . U. Condon and Ⅱ . P. Robertson for their own introduction tO quantum mechanics. The adVice of Professor R. C. ToIman is gratefully acknowledged, as well as the aid of Professor P. M. Morse, Dr. L. 刊 . Sutton, Dr. G. W. Wheland, Dr. L. O. Brockway, Dr. J. Sherman, Dr. S. Weinbaum, Mrs. Emily Buckingham Wilson, and Mrs. Ava HeIen PauIing. LINUS PAULING. E. BRIGHT WILSON, JR. PASADENA, CALIF. , CAMBRIDGE, MASS. , ま リ , 1935.

24 SURVEY 0 CLASSICAL ユ ー 刃 C 〃 Ⅳ CS [I -4 the introduction Of the kinetic and potential energies. BY defining the Lagrangian function for the special case 0f Newtonian systems and introducing it intO the equations 0f motion, Newton's equations were then thrown intO the Lagrangian form. F0110w- ing an introductory discussion 0f generalized coordinates, the proof 0f the general validity 0f the equations 0f motion ⅲ the Lagrangian form for any system 0f coordinates has been given ; and it has also been pointed out that the Lagrangian form of the equations 0f motion, although we have derived it from the equations 0f Newton, is really more widely applicable than Newton's postulates, because by making a suitable ch0ice 0f the Lagrangian function a very wide range 0f problems can bc treat ed in this way. ln the second section there has been derived a third form for the equations 0f motion, the Hamiltonian form' following the introduction 0f the concept 0f generalized momenta, and the rela- tion between the Hamiltonian function and the energy has been discussed. ln S e ction 3 a very brief discussion 0f the classical theory 0f the radiation of energy from accelerated charged particles has been given, in order tO have a foundation for 0f this topic. Mention is made 0f b0th dipole and quadrupole radiat ion. Finally, several examples (which are later solved by the use 0f quantum mechanics) , including the three-dimensional harmomc oscillator ⅱ 1 cartesian and in polar coordinates, been treated by the methods discussed ⅲ this chapter. Genera1 References on C! 、 assical Mechanics W. D. MACMILLAN•. "Theoretical Mechanics. Statics and the Dynamics of a ParticIe," McGraw-HiII Book Company, lnc. , New York' 1932. S. L. LONEY: " Dynamics of a Particle and of Rigid B0dies"' Cambridge Universit,y Press, Cambridge, 1923. 工 H. JEANs: “ TheoreticaI Mechanics, ” Ginn and CompanY' 1 開 7. . T. WHITTAKER: "Analytical Dynamics," Cambridge UniversitY Press, Cambridge, 1928. R. C. TOLMAN: “ Statistical Mechanics with Applications tO Physics and Chemistry," Chemical CataIog Company, lnc. , New York' 1927 , Chap. Ⅱ , The EIements of Classical Mechanics. W. . BYERLY : “ Generalized Coordinates, ” Ginn and CompanY' Bost011' 1916.