INTRODUCTORY QUANTUM MECHANICS Richard L. Liboff Cornell University .A 'Y'Y ADDISON-WESLEY PUBLISHING COMPANY Reading, Massachusetts· Menlo Park, California· New York Don Mills, Ontario· Wokingham, England· Amsterdam Bonn· Sydney· Singapore. Tokyo· Madrid Bogota · Santiago · San Juan INTRODUCTORY QUANTUM MECHANICS Previously published by Holden-Day, Inc. Copyright© 1980 by Addison-Wesley Publishing Company, Inc.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Published simultaneously in Canada.
ISBN 0-201-12221-9 ABCDEFGHIJ-HA-8987 PREFACE This work has emerged from an undergraduate course in quantum mechanics which I have taught for the past number of years. The material divides naturally into two major components. In Part I, Chapters 1 to 8, fundamental concepts are developed and these are applied to problems predominantly in one dimension. In Part II, Chapters 9 to 14, further development of the theory is pursued together with applications to problems in three dimensions.
Part I begins with a review of elements of classical mechanics which are important to a firm understanding of quantum mechanics. The second chapter continues with a historical review of the early experiments and theories of quan- tum mechanics. The postulates of quantum mechanics are presented in Chapter 3 together with development of mathematical notions contained in the statements of these postulates. The time-dependent Schrodinger equation emerges in this chapter.
Solutions to the elementary problems of a free particle and that of a particle in a one-dimensional box are employed in Chapter 4 in the descriptions of Hilbert space and Hermitian operators. These abstract mathematical notions are de- scribed in geometrical language which I have found in most instances to be easily understood by students. The cornerstone of this introductory material is the superposition principle, described in Chapter 5. In this principle the student comes to grips with the inherent dissimilarity between classical and quantum mechanics.
Commutation relations and their relation to the uncertainty principle are also described, as well as the concept of a complete set of commuting observables. Quantum conserva- tion principles are presented in Chapter 6. Applications to important problems in one dimension are given in Chapters 7 and 8. Creation and annihilation operators are introduced in algebraic construction of the eigenstates of a harmonic oscillator.
Transmission and reflection coeffi- cients are obtained for one-dimensional barrier problems. Chapter 8 is devoted primarily to the problem of a particle in a periodic potential. The band structure of the energy spectrum for this configuration is obtained and related to the theory of electrical conduction in solids. Part II begins with a quantum mechanical description of angular momentum.
viii PREFACE Fundamental commutator relations between the Cartesian components of angular momentum serve to generate eigenvalues. These commutator relations further indicate compatibility between the square of total angular momentum and only one of its Cartesian components. It is through these commutator relations that a dis- tinction between spin and orbital angular momentum emerges. Properties of angu- lar momentum developed in this chapter are reemployed throughout the text.
In Chapter 10 the Schrodinger equation for a particle moving in three dimen- sions is analyzed and applied to the examples of a free particle, a charged particle in a magnetic field, and the hydrogen atom. In Chapter 11 the theory of representations and elements of matrix mechanics are developed for the purpose of obtaining a more complete description of spin angular momentum. A host of problems involving a spinning electron in a magnetic field are presented. The theory of the density matrix is developed and applied to a beam of spinning electrons.
In Chapter 12 preceding formalisms are employed in conjunction with the Pauli principle, in the analysis of some basic problems in atomic and molecular physics. Also included in this chapter are brief descriptions of the quantum models for superconductivity and superfluidity. Perturbation theory is developed in Chapter 13. Among the many applica- tions included is that of the problem of a particle in a periodic potential, consid- ered previously in Chapter 8.
Harmonic perturbation theory is applied in Ein- stein's derivation of the Planck radiation formula and the theory of the laser. The text concludes with a brief chapter devoted to an elementary description of the quantum theory of scattering. Problems abound throughout the text, and many of them include solutions. Figures are also plentiful and hopefully lend to the instructional quality of the writing.
A small introductory paragraph precedes each chapter and serves to knit the material together. A list of symbols appears before the appendixes. Interspersed throughout the text, especially in the problems, one finds con- cepts from other disciplines with which the student is assumed to have some familiarity. These include, for example: dynamics, thermodynamics, elementary relativity, and electrodynamics.
This policy follows the spirit of one of my cherished late professors, Hartmut Kalman: ·'Physics is not a sausage that one cuts into little pieces." I trust that a mastery of the concepts and their applications as presented in this work will form a solid foundation on which to build a more complete study of quantum mechanics. Many individuals have been helpful in the preparation of this text. I remain indebted to these kind, patient, and well-informed colleagues: D. Fine, PREFACE ix R.
Sincere gratitude is ex- tended to my publisher, Frederick H. Murphy, for his undaunted patience and confidence in this work. During visits at the Universite Libre de Bruxelles and later at the Universite de Paris XI-Centre d 'Orsay, I was able to work on material related to this text. I am extremely grateful to Professor I.
Prigogine and Professor J. Delcroix for the intellectual freedom accorded me during these occasions. LIBOFF CONTENTS Preface vii PART I ELEMENTARY PRINCIPLES AND APPLICATIONS TO PROBLEMS IN ONE DIMENSION 1 Chapter 1 Review of Concepts of Classical Mechanics 3 1.1 Generalized or ''Good'' Coordinates 3 1.2 Energy, the Hamiltonian, and Angular Momentum 6 1.3 The State of a System 19 1.4 Properties of the One-Dimensional Potential Function 24 Chapter 2 Historical Review: Experiments and Theories 28 2.2 The Work of Planck.3 The Work of Einstein. The Photoelectric Effect 34 2.4 The Work of Bohr.
A Quantum Theory of Atomic States 38 2.5 Waves versus Particles 41 2.6 The de Broglie Hypothesis and the Davisson-Germer Experiment 44 2. 7 The Work of Heisenberg. Uncertainty as a Cornerstone of Natural Law 51 2.8 The Work of Born.9 Semiphilosophical Epilogue to Chapter 2 55 Chapter 3 The Postulates of Quantum Mechanics. Operators, Eigenfunctions, and Eigenvalues 64 3.1 Observables and Operators 64 3.2 Measurement in Quantum Mechanics 70 3.3 The State Function and Expectation Values 73 3.4 Time Development of the State Function 77 3.5 Solution to the Initial- Value Problem in Quantum Mechanics 81 xii CONTENTS Chapter 4 Preparatory Concepts.
Function Spaces and Hermitian Operators 86 4.1 Particle in a Box and Further Remarks on Normalization 86 4.2 The Bohr Correspondence Principle 91 4.6 Properties of Hermitian Operators 104 Chapter 5 Superposition and Compatible Observables 109 5.1 The Superposition Principle 109 5.2 Commutator Relations in Quantum Mechanics 124 5.3 More on the Commutator Theorem 131 5.4 Commutator Relations and the Uncertainty Principle 134 5.5 ·'Complete" Sets of Commuting Observables 137 Chapter 6 Time Development, Conservation Theorems, and Parity 143 6.1 Time Development of State Functions 143 6.2 Time Development of Expectation Values 159 6.3 Conservation of Energy, Linear and Angular Momentum 163 6.4 Conservation of Parity 167 Chapter 7 Additional One-Dimensional Problems. Bound and Unbound States 176 7 .I General Properties of the One-Dimensional Schrodinger Equation 176 7.2 The Harmonic Oscillator 179 7.3 Eigenfunctions of the Harmonic Oscillator Hamiltonian 187 7.4 The Harmonic Oscillator in Momentum Space 199 7.6 One- Dimensional Barrier Problems 211 7.7 The Rectangular Barrier.8 The Ramsauer Effect 224 7.9 Kinetic Properties of a Wave Packet Scattered from a Potential Barrier 230 7 10 The WKB Approximation 232 CONTENTS xiii Chapter 8 Finite Potential Well, Periodic Lattice, and Some Simple Problems with Two Degrees of Freedom 256 8.1 The Finite Potential Well 256 8.3 Standing Waves at the Band Edges 284 8.4 Brief Qualitative Description of the Theory of Conduction in Solids 291 8.5 Two Beads on a Wire and a Particle in a Two- Dimensional Box 294 8.6 Two- Dimensional Harmonic Oscillator 300 PART II FURTHER DEVELOPMENT OF THE THEORY AND APPLICATIONS TO PROBLEMS IN THREE DIMENSIONS 307 Chapter 9 Angular Momentum 309 9.2 Eigenvalues of the Angular Momentum Operators 318 9.3 Eigenfunctions of the Orbital Angular Momentum Operators i 2 and iz 326 9.4 Addition of Angular Momentum 345 9.5 Total Angular Momentum for Two or More Electrons 353 Chapter 10 Problems in Three Dimensions 359 10.1 The Free Particle in Cartesian Coordinates 359 10.2 The Free Particle in Spherical Coordinates 365 10.3 The Free-Particle Radial Wavefunction 370 10.4 A Charged Particle in a Magnetic Field 380 10.5 The Two-Particle Problem 383 10.6 The Hydrogen Atom 394 10.7 Elementary Theory of Radiation 410 Chapter 11 Elements of Matrix Mechanics.1 Basis and Representations 418 11.2 Elementary Matrix Properties 426 11.3 Unitary and Similarity Transformations in Quantum Mechanics 430 11.4 The Energy Representation 436 11.5 Angular Momentum Matrices 442 xiv CONTENTS 11.6 The Pauli Spin Matrices 450 11.7 Free-Particle Wavefunctions, Including Spin 455 11.8 The Magnetic Moment of an Electron 457 11.9 Precession of an Electron in a Magnetic Field 465 11.10 The Addition of Two Spins 474 11. 11 The Density Matrix 481 Chapter 12 Application to Atomic and Molecular Physics. Elements of Quantum Statistics 491 12.1 The Total Angular Momentum, J 491 12.2 One-Electron Atoms 496 12.3 The Pauli Principle 508 12.4 The Periodic Table 514 12.5 The Slater Determinant 520 12.6 Application of Symmetrization Rules to the Helium Atom 523 12.7 The Hydrogen and Deuterium Molecule 532 12.8 Brief Description of Quantum Models for Superconductivity and Superfluidity 539 Chapter 13 Perturbation Theory 549 13.1 Time-Independent, Nondegenerate Perturbation Theory 549 13.2 Time-Independent, Degenerate Perturbation Theory 560 13.3 The Stark Effect 568 13.4 The Nearly Free Electron Model 571 13.5 Time-Dependent Perturbation Theory 576 13.7 Application of Harmonic Perturbation Theory 585 13.8 Selective Perturbations in Time 594 Chapter 14 Scattering in Three Dimensions 605 14.3 Center-of-Mass Frame 617 14.4 The Born Approximation 621 List of Symbols 627 CONTENTS XV Appendixes 631 A Additional Remarks on the x and p Representations 633 B Spin and Statistics 637 C Representations of the Delta Function 639 D Physical Constants and Equivalence (.) Relations 642 Index 645 PART I ELEMENTARY PRINCIPLES AND APPLICATIONS TO PROBLEMS IN ONE DIMENSION CHAPTER 1 REVIEW OF CONCEPTS OF CLASSICAL MECHANICS 1.1 Generalized or "Good" Coordinates 1.2 Energy, the Hamiltonian, and An;?ular Momentum 1.3 The State of a System 1.4 Properties of the One-Dimensional Potential Function This is a preparatory chapter in which we review fundamental concepts of classical mechanics important to the development and understanding of quantum mechanics.
Hamilton's equations are introduced and the relevance of cyclic coordinates and con- stants of the motion is noted. In discussing the state of a system, we briefly encounter our first distinction between classical and quantum descriptions. The notions of forbidden domains and turning points relevant to classical motion, which 'find application in quantum mechanics as well, are also described. The experimental motivation and historical back- ground of quantum mechanics are described in Chapter 2.1 GENERALIZED OR "GOOD" COORDINATES Our discussion begins with the concept of generalized or good coordinates.
A bead (idealized to a point particle) constrained to move on a straight rigid wire has one degree of freedom (Fig. This means that only one variable (or parameter) is needed to uniquely specify the location of the bead in space. For the problem under discussion, the variable may be displacement from an arbitrary but specified origin along the wire. 4 REVIEW OF CONCEPTS OF CLASSICAL MECHANICS x=O X FIGURE 1.1 A bead constrained to move on a straight wire has one degree of freedom.