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Richard L. Liboff
INTRODUCTORY
QUANTUM MECHANICS
Richard L. Liboff
Cornell University
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ADDISON:
YESLEY PUBLISHING COMPANY
Reading, Massachusetts: Meulo Park, California-New York
Don Mills, Ontario- Wokingham, England- Amsterdam
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INTRONUCTORY QUANTUM MECHANICS
Previously published by Holden-Day, Inc,
Copyright 1980 by Addison-Wesley Publishing Company, Ie
[All sights ronerved, No part of this publication may be reproduc
Stored in a retticyal system, vt transmitted, in any form of by ary
tneans, electronic, mechanical, photocopying, recording, or othetwisc,
without the prior sritten permission of the publisher.
Printed in the United Staces of Ameriea.
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
inte two major components. In Part 1, Chapters 1 fo 8, fndamental concepts are
developed and these are applied to problems predominantly in one dimension, In
Part 11, Chapters 9 to 14, further development of the theory is pursued together
with applications to problems in three dimens
Part [ begins with a review of elements of classical mechanics which are
important to a firm understanding of quantum mechanics. Lhe second chapter
continues with a historical review of the early experiments and theoties of quan-
tum mechanics. The postulates of quantum mechanics are presented in Chapter 3
together with development of mathematical notions contained in tke statements of
these postulates, The time-dependent Schridinger equation emerges in this
chapter.
Solutions to the elementary problems of a free particle and thal of a parti
in @ one-dimensional box are employed in Chupter 4 in the descriptions of Hilbert
space and Hermitian operators. These abstract mathematical nouions are de-
soribed in geometricul 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, Tn this principle the student comes to grips with the
inherent dissimilarity between classical and quantum mechatics, Commutation
relations and their relation to the uncertainty principle are also described, as well
ay the concept of a complete set of commuting observables, Quantuzn comer
ples 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 cocff-
cients are obtained for one-dimensional barrier problems. Chapter 8 is devoted
primarily to the problem of a parliele 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
Purt Il begins with a quantum mechanical description of angular momentum.
on
Fundamental commutator relations between the Cartesian components of angular
momentum serve to generate cigenvalues. These commutator relations further
indivate compatibility between the square of tota angular momentum and oaly one
of ils Cartesian components. It is through these commutator relations that i dis
tinction between spin and orbital angular momentum emerges, Properties of angu-
lar momertum developed in this chapter are reemployed throughout the text
In Chapter 10 the Schrédinger equation for a particle moving in three uimen-
sions is unalyzed and applied to the examples of a free particle, u charged particle
in 4 magnetic field, and the hydrogen atom.
In Chapter 11 the theory of representations und elements of matrix
mechanics are eveloped 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
Puuli 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 patticle in a periodic potential, consid.
ered previously in Chapter 8, Harmonic perturhation theory is applied in Kin-
stem’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 (heory 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 smnall introductory paragraph precedes each chapter and serves to knit
the matcrial together. A fist 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 t@ 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 tittle pieces.”
Ltrusl that a mastery of the convepls and their applications as prevented in
this work will form a solid foundation on which lo build 2 more complete study of
quantum mechanics
Many individuats have been helpful in the prepuration of this text. I remain
indebted to these kind, patient, and well-informed collegues: D. Heffernan, M
Guillen, E, Dorchak, D. Faulconer, G. Lasher, I. Nebenzahl, M. Nelkin, F, Fine.
R. McFarlane, C. Tang, K. Gottfried, and G, Severe. Sincere gratitude is exe
tended to my publisher, Frederick H. Murphy, for his undaunted patience and.
confidence in this work,
During visits at the Université Libre de Bruneltes and later at dhe Université
de Paris XI-Centre d’Orsay, [was able to work on material retated to this text. 1
am extremely grateful to Professor 1. Prigogine and Professor I. L. Deleroix for
the intellectual freedom accorded me during these occasions
RT, TAROFF
Preface
pare
Chapter 1
Chapter 2
Chapter 3
CONTENTS
ELEMENTARY PRINCIPLES AND APPLICATIONS TO
PROBLEMS IN ONE DIMENSION
Review of Concepts of Classical Mechat
L1 Generalized or Good" Coordinates
12 Energy, the Hamiltonian, and Angular Momentum
1.3. The State of a System
1.4 Properties of the Ong-Dimensional Potential Function
Historical Review: Experiments and Theories
21 Dates
2.2, ‘The Work of Planck. Blackbody Radiation
2.3 The Work of Einstein. The Photoelectric Effect
2.4. The Work of Bohr. A Quantum Theory of Atomic States
28 Waves versus Particles
2.6 The de Broglie Hypothesis and the Davisson-Germer
Experiment
The Work of Heisenberg. Uncertainty as a Cornerstone
of Natural Law
2.8 The Work of Born, Probability Waves
2.4 Semiphilosophical Epilogue to Chapter 2
The Postulates of Quantum Mechanies. Operators,
Figenfunctions, and Eigenvalues
3.1 Observables and Operators
3.2, Measurement in Quantum Mechanics
2.3 The State Function and Expectation Values
3.4 Time Development of the State Function
3.5 Solution lo the Initial Value Problem in Quantum
Mechanics
all covrenty
Chapter 4
Chapter §
Chapter 6
Chapter 7
Prepuratory Concepts. Function Spaces and Hermitian
Operators
4.1 Particle ina Box and Further Remarks on
Normalization
4.2, The Bohr Correspondence Principle
4.3 Direc Notation
4.4 Hilbert Space
4.8 Hermitian Operators
4.6 Properties of Hermitian Operators
Superposition and Compatible Observables
5.1. The Superposition Principle
5.2. Commutator Relations in Quantum Mechanics
5.2. More on the Commutator Theorem
5.4 Commutator Relations and the Uncertainty Principle
5.5 ~Comptete’’ Sets of Commuting Observables
‘Time Deselopmeat, Conservation Theorems, and Parity
6.1. Time Development of State Functions.
6.2. Time Development of Expectation Values
6.2 Conservation of Energy, Linear and Angular
Momentum
6.4 Conservation of Parity
Additional One-Dimensional Problems. Bound und
Unbound States
7.1. General Properties of the One-Dimensional
Schrodinger Equation
7.2. The Harmonie Oscillator
7.3 Figenfunctions of the Harmonic Oscillator
Hamiltonian
4 The Harmonic Oscillator in Momentum Space
5 Unbound States
6 One-Dimensional Barrier Problems
7 The Rectangular Barrier, Tunneling
R The Ramsauer Effect
7.9 Kinetic Properties of a Wave Packet Scattered from a
Potential Barrier
7.10 The WKB Approximation
36
86
a
8
94
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104
109
109
124
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143
159
163
167
76
Chapter § Finite Potential Well, Periodic Lattice, and Some Simple
Eroblems with Two Degrees of Freedom 256
8.1 The Finite Potential Well 2
8.2 Periodic Lattice. Energy Gaps 267
8.3. Standing Waves at (he Band Edges 28a
8.4 Brief Qualitative Description of the Theory of,
Conduction in Solids 291
8.5 Two Beads ona Wire and a Particle in a
Two-Dimensional Box 294
8.6 Two-Dimensional Harmonic Oscillator 300
vakr Il FURTHER DEVELOPMENT OF THE THEORY AND
APPLICATIONS TO PROBLEMS IN THREE,
DIMENSIONS 307
Chapter 9 Angular Momentum 309
9.4 Basic Properties 310
9.2 walues of the Angular Munsentum Operators 318
9.3. Eigenfunctions of the Orbital Angular Momentum
Operators £¢ and f., 326
9.4 Addition of Angulat Momentum 345
9.5 Total Angular Momentum for Two or More Flectrons 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 37
10.4 A Charged Particle in a Magnetic Field 380
10.5. The Two-Perticle Problem 383
1.6 The Hydrogen Atom 394
10,7 Elementary Theory of Rediation 4a)
Chapter 11 Elements of Matrix Mechanics. Spin Wavefunctions 418
11.1. Basis and Representations ag
11.2 Elementary Matrix Properties 426
11.3. Unitary and Similarity Transformations in Quantum
Mechanics 430
114 The Energy Representation 36
1.5. Angular Momentum Matrices a
My CONTENTS
Chapter 12
Chapter 13
Chapter 14
List of
Symbols
11.6 The Pauli Spin Matrices
117 Free-Particle Wavefunetions, Including Spin
11.8 The Magnetic Moment of an Electron
11.9 Precession of an Electron in a Magnetic Field
11.10 The Addition of Two Spins
11.11 The Density Matrix
Application to Atomic and Molecular Physics. Elements of
Quantum Statistics
12.1. The Total Angular Momentum, J
12.2. One-Electron Atoms
12,3 The Pauli Principle
12.4 The Periadie Table
12.5 The Slater Determinant
12.6 Application of Symmetrization Rules to the Helium
Alom
12.7 The Hydrogen and Deuterium Molecule
12.8 Brief Description of Quantum Models for
Superconductivity and Superfiuidity
Perturbation Theory
13.1 ‘Time-Independent, Nondegenerate Perturbation
Theory
13.2. Time-Independent, Degenerate Perturbation Theory
13.3 The Stark Effect
13.4. The Nearly Free Electron Model
13.5. Time-Dependent Perturbation Theory
(3.6 Harmonic Perturbation
13.7 Application of Harmonic Perturbation Theory
12.8. Selective Perturbations in Time
Scattering in Three Dimensions
14.1 Partial Waves
14.2, 5-Wave Scattering
14.3 Center-of-Mass Frame
14.4 The Born Approximation
627
