INTERNATIONAL SERIES IN PURE AND APPLIED PHYSICS G. P. HARNWELL, Consuttive Eprron Apvisory Eprroria, Commrrres: E, U. Condon, George R. Harrison, Elmer J[utehisson, K. K. Darrow QUANTUM MECHANICS Donated by Mrs. Yemuna Bappu to The Indian Institute of Astrophysics trom the personal collection of Dr. M. K. V. Bappu INTERNATIONAL SERIES IN PURE AND APPLIED PHYSICS G. P. Harnwuun, Consulting Editor Bacwea ann Goupsszr—ATOMIC ENERGY STATES Brrraz—INTRODUCTION TO FERROMAGNETISM Bririowx—WAVE PROPAGATION IN PERIODIC STRUCTURES Oso¥—PIELORLECTRICITY Crank—APPLIED X-RAYS Cunmis—ELECTRICAL MEASUREMENTS Daver—CRYSTAL STRUCTURE AND ITS APPLICATIONS Epwanps—-ANALYTIC AND VECTOR MECHANICS Hanpy an Puurin—THE PRINCIPLES OF OPTICS Harwwe1—ELECTRICITY AND ELECTROMAGNETISM Hagnwawt aNp Lryincooo EXPERIMENTAL ATOMIC PHYSICS Hoveron—PRINCIPLES OF MATHEMATICAL PHYSICS Hoamns ano DuBatwor—PHOTOELECTRIC PHENOMENA Hunp—HIGH-FREQUENCY MEASUREMENTS PHENOMENA IN HIGH-FREQUENCY SYSTEMS Inoprsot, Zonet, anp Inomesont—HEAT CONDUCTION Kewein—PRINOIPLES OF QUANTUM MECHANICS Kannarn—KINETIC THEORY OF GASES Kousr—THE PHYSICS OF ELECTRON TUBES Morse—VIBRATION AND SOUND Pavuine ano Govpenat—THE STRUCTURE OF LINE SPHCTRA Rucwrmyer avn Kenwarn—INTRODUCTION TO MODERN PHYSICS Rosa ano Urer—ATOMS, MOLECULES, AND QUANTA Soure—QUANTUM MECHANICS Sarm—TH MODURN THEORY OF SOLIDS Statze—INTRODUCTION TO CHEMICAL PHYSICS MICROWAVE TRANSMISSION Sarr any Franx—ELECTROMAGNETISM INTRODUCTION TO THEORETICAL PHYSICS MECHANICS Suyruz—STATIC AND DYNAMIC ELECTRICITY Srrartox—LLECTROMAGNETIC THEORY Wurrs—INTRODUCTION TO ATOMIC SPECTRA ‘Wirztams—MAGNETIC PHENOMENA Dr, Lee A, DuBridge was consulting editor of the series from 1939 to 1946. QUANTUM MECHANICS BY LEONARD 1, SCHIFF Professor of Physics Stanford University First Hoveron Sxconp IMpiession: AW YORK TOROVrO LONDON McGRAW-LILL BOOK COMPANY, INC. 1949 uA Lib. TA ARE g QUANTUM MECHANICS Copyright, 1949, by the McGraw-Hill Book Company, Ine. Printed in the United States of America, All rights reserved. This book, or purts thereof, gaay not be reproduced in any form without permission of the publishers. ‘THE MAPIE PRESS COMPANY, YORK, PA. PREFACE This volume has a threefold purpose: to explain the physical concepts of quantum mechanics, to describe the mathematical formalism, and to present illustrative examples of both the ideas and the methods. The book is intended to serve as a text at the graduate level and also as a reference book. It is assumed that the reader is reasonably familiar with atomic structure, classical mechanics, and differential equations. Tn addition, he should have had some contact with electromagnetic theory and, for the latter part of the book, with the special theory of relativity. The author believes that the analytical methods employed in the book will satisfy most theoretical physicists even though no attempt is made to achieve mathematical rigor. For example, there is little or no dis- cussion of the justification for the interchange of sum, derivative, and. integral operations, or for the usc of the 6 function. On the other hand, the physical reasons for the nature of the results obtained are investigated. wherever possible. Problems are given nt the end of each chapter. They are often used to illustrate or amplify points discussed in the text. Original theorctical papers are referred to throughout the hook; the list is representative rather than exhaustive. Experimental results are, for the most part, quoted without reference, since the large amount of documentation required for an adequate survey seems out of place in a book on theoretical physics. Several othor books on quantum mechanics and related sub- jects arc referred to for more detailed discussions of particular topics. ‘The scope of this volume is best outlined if the book is divided into three parts. The first three chapters constitute an introduction to quantum mechanies, in which the physical concepts are discussed and the Schrédinger wave formalism is established. The detniled treatment of the wave function (Chap. TH) may be omitted ina first rending. The next eight chapters comprise the central part of the book. This part presents exact solutions of the wave equation for both energy-level and collision problems, the Heixenberg matrix formalism and transformation theory, approximation methods, radiation theory, and some applications to atomic sysiems. Since the first eleven chapters correspond to a typical one-year graduate course, il seemed desirable to inelude a semi- classical treatment of clectromagnetic radintion in the contral part of the hook (Chap. X) even though some of the results are obtained again in Chap. XIV. The last part of the book corresponds to a short course in what is often called advanced quantum mechanicr, It consists of rela- v PREFACE cle theory and an introduction to quantized field theory and aectrodynamics. all this book was written while the author was at the Uni- ut Pennsylvania, and he gratefully acknowledges the continued ugement of Prof. G. P. Harnwell. He is also indebted 10 Drs. E. EH. .....ard and §, Pasternack for helpful criticism of the early chapters, to W. Miller and L. Spruch for their careful reading of most of the manu- script, and to Profs. F. Bloch, R. F. Christy, and W. W. Hunsen for valuable comments on certain sections. It is a particular pleasure for the author to thank Prof. R. Serber for many discussions of both the conceptual and formal aspects of quantum mechanics that took place during the Jast eleven years. Finally, the author wishes to acknowledge his indebtedness to Prof. J. R. Oppenheimer for his introduction to severul of the ideas and exam- ples that appear in the book. Indeed, the writing of this book in its present form owes much to the period from 1937 to 1840, which the author spent in association with Professor Oppenheimer. Lxuonann I, Scrtrr Sranvorp, CALIFORNIA Fane, 1948 Pre 9. 10, CONTENTS ACD. ee ee Pee Cuarrer T THE PHYSICAL RASIS OF QUANTUM MECHANICS . Haperimental Background. 6. 0 ee Tnadequacy of classical physics. Summury of prineipal experiments and inferonces. The Old Quantum Pheary. . . bee e eee eB Bobr-Sommerfeld quantization rules, Practical difficulties. Conceptual difficulties, Quantum-mechanieal viewpoint. Uncertainty and Complementarity 0 eB Uncertainty principle. Complementarity principle. Lintitati periment. Discussion of Measurement. ee DO Anulysis of diffraction experiment. Discussion of diffrs Tocatiaition experiment, Wave Packets in Spnee and Space packets. Time p tion experiment. a re P ots. Wave f Cuaeren IL THE SCHRODINGER WAVE EQUATION Denlopmient of the Wane Bquation .. . . fee WT Traveling harmonie waves. Need for a wave entiation, ‘The owedimen= sion) wave equation. Extension to three dimensions. Inclusion of for Taterpretation of the Wann Punetion. . . . . . . oe al Stalintieal interpretation, Normalisation of yj. Peohabll sity. Expectation value, Khrenfost'y theorem, Energy Bigenfunction® 0 .. Seperation of the wave equation. Significance of the separation co stunt B. Boundary conditions at great distanes, Continuity conditions, Boundary conditions for infinite potential energy. Energy eigenvalues one dimension, Discrete energy levels. Continuous energy cigenvalucs. Diserete and continuous cigenvilues in Uhees dimensions. One-Dimensional Syuare Well Potentinl. 6. ee vb Perlectly rigid walls, Finite: potential step. Energy levels. Pac simplified solrtion. y current den in t ITT TWIGENEFUNCTIONS AND BLIGENVALITES Frterpretative Postitates and Energy Kigenfunctions. we Al Dymamieal variables as operators, Expansion in eigenfunctions, The vi ie 12. 13. id. 16. 17, 18. 19. 20, . Three-Dimensional Square Well Potential CONTENTS total energy operator. Normalization in a box. Orthonormality of energy eigenfunctions. Reality of the energy eigenvalues. Expansion in energy cigenfunctions. The closure property. Probability function and expectation value. General solution of the Sehridinger equation, Momentum. Eigenfunctions . . . Form of the eigenfunetions. Box normalization, ‘The Dirac § function. A representation of the 5 function, Normalization in terms of the 3 fno- tion, Some properties of the 5 function, Closure. Expansion in momentum eigenfunctions, Probability function and expectation value. Motion of a Free Wave Packet in One Dimension, : ‘The minimum uncertainty product. Form of the minimum packet. Momentum expansion coefficients, Change with time of a minimum packet. Classical limit. Caprese IV DISCRETE EIGENVALUES: ENERGY LEVELS Linear Harmonie Oscillator © be Asymptotic behavior. Energylovels. Zero-point energy. Parity. Here mite polynomials, Harmonic-oseillator wave functions. Correspondence with classical theory, Oseillating wave packet. Spherically Symmetric Potentials in Three Dimensions... a Separation of the wave equation. Legendre polynomials, Sphe harmonies, Parity. Angular momentum, Zero angular momentum, Interior solutions for arbitrary f, Exterior solutions for arbitrary 1. Energy levels. The Hydrogen Atom ee Redneod mass. Asymptotic behavior. Energy levels. Laguerre_ pol nomials, Tlydrogen-atom wave functions. TDogeneracy. Separation i parabolic coordinates. Energy lovels, Wave funetions. Cuaprar V CONTINUOUS TFIGENVALURS: COLLISION TITKORY One-Dimensional Square Potential Barrier... . Asymptotic behavior. Normalization. Scattering Coeffieients. Collisions in Three Dimensions... Scattering cross section, Relations between angles in the laborntory and conter-of-mass systems, Relation between cross sections. Dependence ony. Asymptotic behavior. Normalization. Scattering by Spherically Symmeiric Potentials Lee a Asymptotic behavior. Scattering cross section, Phase shift, Caleu- lation of 8. Relation between signs of and V(r). Ramsuuer-Townsend effect. Scattering hy a perfoctly rigid sphere. Scattering by a square well potential. Resonance stattering. Angular distribution at low energies. Scattering by a Coulomb Field . : an Parabolic coordinates, Confluent hypergeometric function, Seattering cross section and normalization. Solution in sphorieal coordinates. Modi- fied Coulomb field. Classical limit for a pure Coulomb field. 48 Bt GO 69 76 80 . 103 meecy Ql. 22. 23, 24, 28. . Stationary Perturbation Theory. CONTENTS Cuarrer VI MATRIX FORMULATION OF QUANTUM MECHANICS Matrix Algebra. . Lee Matrix addition and multiplication. Null, unit, and constant matrices. Spur, determinant, and inverse of a matrix, Hormitian end unitary matrices. ‘Transformation and diagonalization of matrices, Matrices of infinite rank. Matrices in Quantum Mechanics. Lee eee A unitary transformation matrix. Hamiltonian matrix, Dynamical variables as Hermitian matrices. Wave functions as unitary matrices. The z representation. A useful identity. Equations of Motion in Matrix Form. Le eee ‘Time rate of change of matrix. Classical Lagrangian and Hamiltonian equations of motion. Poisson brackets and commutator brackets. Quan- tization of a classical system. Motion of a particle in an electromagnetic field. Evaluation of commutator brackets. Velocity and acceleration of a charged particle. The Lorenta force. Constants of the motion, The parity operator. Energy representation. Angular Momentum . . Lee ee Definition of angular momentum. Choise of a representation, Relations between matrix clements, Kigenvalues of MW. Eigenvalues of M3; the Linateix. Connection with tho spherical harmonies, Spin angular mo- mentum. Addition of angular momenta. Kigenvalues of (ML, + Mz)% Car VIL APPROXIMATION METHODS FOR STATIONARY PROBLEMS: nid-order perturba~ tor, Degenerate Nondegenerate case, First-order perturbation, See tion. Normal nol y. Perturbation of an oxe case. Removal of degeneracy in first order, Removal of degeneracy in second order. Vinst-order Stark effect in hydrogen. Perturbed energy levels. Ovcurrence of permanent cloelric-dipole moments. The Born Approximation. . Perturbation approximation, free particle, Kxplieit e tion treatment ofp: well potential, Vali¢ serconed Coulomb field. The Variation Method we : re woe Txpeetation vw Fihe ouergy. Application to exeited stat Ground stateal helium. Kleetron interaction energy. Variation of the parameter Z. yan der Waals interaction. Perturbation calculation. Variation cal- culatio The WKB Approximation Le Classical limit, Approximate solutions, Asymptotie nature of the solu- tions. Solutinn near a turning point, Linear turning point. Connection at the turning point. Asymptotic connection formulas, Energy levels of a potential well. A quantization rule. Special houndary conditions. function. Clroen’s function for a Ination, Scattering eros section. Perturba- waves. Phaso shifte. Seuttering hy a square iy of the Born approximation. Seattering by a ix . 22 . 127 » 131 . 140 . 149 . 159 . 169 . IR x CONTENTS Cnarren VIIT APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS 29, Time-Dependent Perturbation Theory . Fe 189 Expansion in unperturbed eigenfunctions. ¥ storder perturbation, Physical interpretetion, ‘Transition probability. Scattering cross xeetion, Harmonie perturbation. Second-order perturbation, Vffect, af first-order transitions, Intermediate states. 30. Inelastis Collisions, 2. 2. ee re) Expression for the cross seotion, Evaluation of the matrix element, Differential and total cross sections. Production of a clowd-chamnber track. Formulation of the problem. Evaluation of the k sum. Second. order matrix element, Discussion of the cross section. 81. Adiabatic and Sudden Approvimations .... 1. . cee ee . 207 Expansion in instantaneous energy eigenfunctions. Choive of phases, Adiabatic approximation. Connection with perturbation theary. Dine. continuous change in H. Sudden approximation. Transient disturbance, Disturbance of an oscillator. Cuarrern IX IDENTICAL PARTICLES AND SPIN 32, Identical Particles... . . ae . Physical meaning of identity. Symmetric and antisyramcti wave f tions. Construction from unsymmetrized functions. Dislingn viity of identical particles. The exclusion principle. Conneetion with stat is: cal mechanics. Collisions of identical particles. 83. Spin Angular Momentum. woe Connection betweon spin and statictios, Spin mut Collisions of identical particles, Electron spin fun: atom. Spin functions for three electrons. 34. Rearrangement Collisions. 2... 0. be .. 230 Notation for rearrangement collisions, Use of the Bori approximation. Lack of orthogonality of initial and final states. Connoetion with time dependent perturbation theory. Exchange collisions of clerirons with hydrogen. Differential cross section, Exchunge collisions with heliuun. ancl cigonfunetionss. ns. The helium Crarrer X SEMICLASSICAL TREATMENT OF RADIATION 85, Absorption and Induced Emission . . ee B10 Maxwell's equations. Plane electromagnetic waves, Uwn of peeturrlat ten theory. Transition probability. Interpretation in forma of absorptions Hostiedipale wansilions, Forbidden transitions, %5. Spontaneous Eméaion. ok an . BAX a energy. Dipole radiation. Angular momentue, Dipole e caso, Conversion from elagsien] ta quantum theory. Planck distribution formula. Line breadth. 87. Some Applications of Radiation Theory . Selection rules for a single particle. Pola 287 ‘ation of emitted radiation.