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Elementary quantum mechanics PDF

438 Pages·1968·55.845 MB·English
by  SaxonDavid S
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ELEMENTARY QUANTUM MECHANICS ELEMENTARY QUANTUM MECHANICS David S. Saxon University o/California, LosAngeles no HOLDEN-DAY San Francisco, Cambridge. London,Amsterdam © Copyright 1968 by Holden-Day, Inc., 500 Sansome Street, San Francisco, California. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. Library ofCongress Catalog Card Number 68-16996. Printed in the United States ofAmerica. Preface This book is based on lectures given by the author in an intensive undergraduate course in quantum mechanics which occupies a central role in the physics curriculum at UCLA. It is a required course for all third-year physics and astrophysics students, but it is taken by some seniors and manygraduate students,both in physics and in related fields. Students enrolling in the course are expected to have hadan introduc tion to elementary Hamiltonian mechanics, to the extent of knowing, for simple systems, what the Hamiltonian function is and what the Hamiltonian equations are. Students are also expected to have had training in mathematics through differential equations and Fouriersenes and to have at least seen many ofthe special functions ofmathematical physics. In an effort to keep the mathematics as simple as possible, however, the first two-thirds of the book is largely confined to the con sideration ofone-dimensional systems. The stress throughout is on the formulation of quantum mechanics and not on its applications. At UCLA the applications follow in immedi ately subsequent courses selected from atomic, nuclear, solid state and elementary particle physics. The lastchapteris intended to pave the way for these applications; in it a number of relatively advanced topics are somewhat briefly presented. The coverage is rather broad and not everything is treated in depth. Wherever the text is frankly introductory, however, references to a complete treatment are given. In all other respects the book is self contained. Experience with a preliminary edition has shown that it is accessible to students and that they can learn from it largely by them selves. To aconsiderabledegree the teacher is thus leftfree to illuminate the subject in his own way. One hundred and fifty problems are presented, and these play an im- vi PREFACE portant pedagogical role. The problems are not exclusively illustrative of material presented in the text; they also amplify it. A significant number are intended to broaden the scope ofthe course by pointing the way to new topics and new points ofview. Many problems are too diffi cult for the student to master in his first attempt. He is encouraged to return to them again and again as his understanding grows. Eventually he should be able to handle any and all of them. Answers or complete solutions to some fifty representativeproblemsaregivenin Appendix Ill. Aboutforty exercises are scattered throughoutthetext. Theseare mostly concerned with the working out ofdetails, but not all ofthem are trivial. At UCLA the materialinthetextis presentedinasequencecovering two quarters. However, the text isalsointendedforuseinaone-semester course; any, or all, of the starred sections in the table of contentS can be omitted without harm to the logical development. Ifit is desired, on the other hand, to use the textfor aone-yearcourse, some supplementa tion would be desirable. The Heisenbergand interaction representations, and transformation theory in general, are topics which at once come to mind. At the applied level, the Zeeman and Stark effects, Bloch waves, the Hartree-Fock and Fermi-Thomas methods, simple molecules and isotopic spin are a suitable list from which to choose. The author has benefited from numerous criticisms and suggestions from a host of colleagues and students. To each of them, he expresses his deep gratitude and especially to Dr. Ronald Blum for his meticulous reading of both the preliminary edition and the final manuscript. The author wi11 be equally grateful for additional comments and for the correction ofmisprints and errors. DavidS. Saxon November, 1967 Contents I. THE DUAL NATURE OF MATTER AND RADIATION I. The breakdown ofclassical physics . I 2. Quantum mechanical concepts. . . . . . . 3 3. The wave aspects of particles. . . . . . . 5 4. Numerical magnitudes and the quantum domain 12 5. The particle aspects ofwaves . 13 6. Complementarity . . . . . 16 7. The correspondence principle. 16 II. STATE FUNCTIONS AND THEIR INTERPRETATION I. The idea ofa state function; superposition ofstates 18 2. Expectation values. . . . . . . . . . . 23 3. Comparison between the classical and quantum descriptions ofa state; wave packets . . . . 25 Ill. LINEAR MOMENTUM I. State functions corresponding to a definite momentum 29 2. Construction ofwave packets by superposition 31 3. Fouriertransforms: the Dirac deltafunction. 34 4. Momentum and configuration space. . 38 5. The momentum and position operators. 39 6. Commutation relations . 45 7. The uncertainty principle . . . . . 47 IV. MOTION OF A FREE PARTICLE I. Motion ofa wave packet; group velocity 56 viii CONTENTS 2. The correspondence principle requirement 59 3. Propagation ofafree particle wave packet in configuration space. 60 4. Propagation ofa free particle wave packet in momentum space; the energy operator . 62 5. Time development ofa Gaussian wave packet 64 6. The free particle Schrodingerequation . 66 7. Conservation ofprobability. 68 8. Dirac bracket notation 72 9. Stationary states 73 10. A particle in a box. 75 II. Summary 81 V. SCHRODINGER'S EQUATION I. The requirement ofconservation ofprobability. 84 2. Hermitian operators 85 3. The correspondence principle requirement 9\ 4. Schrodinger's equation in configuration and momentum space 95 5. Stationary states 97 6. Eigenfunctions and eigenvalues ofHermitian operators 101 7. Simultaneous observables and complete sets ofoperators 104 8. The uncertainty principle 106 *9. Wave packets and their motion 110 10. Summary: The postulates ofquantum mechanics III VI. STATES OF A PARTICLE IN ONE DIMENSION I. General features 117 2. Classification by symmetry; the parity operator. 119 3. Bound states in a square well 121 4. The harmonic oscillator . 127 *5. The creation operator representation 139 *6. Motion ofa wave packet in the harmonic oscillator potential 145 7. Continuum states in a square well potential 147 8. Continuum states in general; the probability flux 153 *9. Passage ofa wave packet through a potential 155 * 10. Numerical solution ofSchrodinger's equation 159 VII. APPROXIMATION METHODS I. The WKB approximation 175 2. The Rayleigh-Ritz approximation 185 3. Stationary state perturbation theory. 189 *For a one-semester course, any or all of the starred sections can be omitted without harm to the logical development(see Preface). CONTENTS ix 4. Matrices . . . . . . . . 201 5. Degenerate or close-lying states 205 6. Time dependent perturbation theory 209 VIII. SYSTEMS OF PARTICLES IN ONE DIMENSION I. Formulation. . . . . . . . . . . . . 227 2. Two particles: Center-of-mass coordinates 229 3. Interacting particles in the presence ofuniform external forces . . . . . . . . . . . . 233 *4. Coupled harmonic oscillators . . . . . . 237 5. Weakly interacting particles in the presence of general external forces . . . . . . . . 239 6. Identical particles and exchange degeneracy. 241 7. Systems oftwo identical particles. . . . . 243 8. Many-particle systems; symmetrization and the Pauli exclusion principle. . . . . . . 245 *9. Systems ofthree identical particles. . . 249 10. Weakly interacting identical particles in the presence ofgeneral external forces 255 IX. MOTION IN THREE DIMENSIONS I. Formulation: Motion ofa free particle. 263 *2. Potentials separable in rectangularcoordinates 265 3. Central potentials; angular momentum states. 269 4. Some examples. . . 279 5. The hydrogenic atom. . . . . . . . . 287 X. ANGULAR MOMENTUM AND SPIN I. Orbital angular momentum operators and commutation relations . . . . . . . 299 2. Angular momentum eigenfunctions and eigenvalues 303 *3. Rotation and translation operators 313 4. Spin: The Pauli operators . . 317 *5. Addition ofangular momentum . 327 XI. SOME APPLICATIONS AND FURTHER GENERALIZATIONS *I. The helium atom; the periodic table. . . . . . .. 345 *2. Theory ofscattering . . . . . . . . . . . .. 351 *3. Green's function for scattering; the Born approximation. 361 *4. Motion in an electromagnetic field . 373 *5. Dirac theory ofthe electron 377 *6. Mixed states and the density matrix. 387 x CONTENTS APPENDICES I. Evaluation ofintegrals containing Gaussian functions. 397 II. Selected references. . . . . . . . . 401 III. Answers and solutions to selected problems . . . . 403 "And now reader,- bestir thyself-for though we will always lend thee proper assistance in difficult places, as we do not, like some others, expect thee to use the arts of divination to discover our meaning, yet we shall not indulge thy laziness where nothing but thy own attention is required; for thou art highly mistaken if thou dost imagine that we intended when we begun this great work to leave thy sagacity nothing to do, or that without sometimes exercising this talent thou wilt be able to travel through our pages with any pleasure or profit to thyself." HENRY FIELDING

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