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Fitts D.D. - Principles of Quantum Mechanics[c] As Applied to Chemistry and Chemical Physics (1999)(en)

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PRINCIPLES OF

QUANTUM

MECHANICS:

as Applied to Chemistry and Chemical Physics

DONALD D. FITTS

CAMBRIDGE UNIVERSITY PRESS

PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

This text presents a rigorous mathematical account of the principles of quantum mechanics, in particular as applied to chemistry and chemical physics. Applications are used as illustrations of the basic theory.

The ®rst two chapters serve as an introduction to quantum theory, although it is assumed that the reader has been exposed to elementary quantum mechanics as part of an undergraduate physical chemistry or atomic physics course. Following a discussion of wave motion leading to SchroÈdinger's wave mechanics, the postulates of quantum mechanics are presented along with the essential mathematical concepts and techniques. The postulates are rigorously applied to the harmonic oscillator, angular momentum, the hydrogen atom, the variation method, perturbation theory, and nuclear motion. Modern theoretical concepts such as hermitian operators, Hilbert space, Dirac notation, and ladder operators are introduced and used throughout.

This advanced text is appropriate for beginning graduate students in chemistry, chemical physics, molecular physics, and materials science.

A native of the state of New Hampshire, Donald Fitts developed an interest in chemistry at the age of eleven. He was awarded an A.B. degree, magna cum laude with highest honors in chemistry, in 1954 from Harvard University and a Ph.D. degree in chemistry in 1957 from Yale University for his theoretical work with John G. Kirkwood. After one-year appointments as a National Science Foundation Postdoctoral Fellow at the Institute for Theoretical Physics, University of Amsterdam, and as a Research Fellow at Yale's Chemistry Department, he joined the faculty of the University of Pennsylvania, rising to the rank of Professor of Chemistry.

In Penn's School of Arts and Sciences, Professor Fitts also served as Acting Dean for one year and as Associate Dean and Director of the Graduate Division for ®fteen years. His sabbatical leaves were spent in Britain as a NATO Senior Science Fellow at Imperial College, London, as an Academic Visitor in Physical Chemistry, University of Oxford, and as a Visiting Fellow at Corpus Christi College, Cambridge.

He is the author of two other books, Nonequilibrium Thermodynamics (1962) and Vector Analysis in Chemistry (1974), and has published research articles on the theory of optical rotation, statistical mechanical theory of transport processes, nonequilibrium thermodynamics, molecular quantum mechanics, theory of liquids, intermolecular forces, and surface phenomena.

PRINCIPLES OF

QUANTUM MECHANICS

as Applied to Chemistry and Chemical Physics

DONALD D. FITTS

University of Pennsylvania

PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING) FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge CB2 IRP

40 West 20th Street, New York, NY 10011-4211, USA

477 Williamstown Road, Port Melbourne, VIC 3207, Australia

http://www.cambridge.org

© D. D. Fitts 1999

This edition © D. D. Fitts 2002

First published in printed format 1999

A catalogue record for the original printed book is available from the British Library and from the Library of Congress Original ISBN 0 521 65124 7 hardback

Original ISBN 0 521 65841 1 paperback

ISBN 0 511 00763 9 virtual (netLibrary Edition)

Contents

Preface

 

viii

Chapter 1

The wave function

1

1.1

Wave motion

2

1.2

Wave packet

8

1.3

Dispersion of a wave packet

15

1.4

Particles and waves

18

1.5

Heisenberg uncertainty principle

21

1.6

Young's double-slit experiment

23

1.7

Stern±Gerlach experiment

26

1.8

Physical interpretation of the wave function

29

 

Problems

34

Chapter 2

SchroÈdinger wave mechanics

36

2.1

The SchroÈdinger equation

36

2.2

The wave function

37

2.3

Expectation values of dynamical quantities

41

2.4

Time-independent SchroÈdinger equation

46

2.5

Particle in a one-dimensional box

48

2.6

Tunneling

53

2.7

Particles in three dimensions

57

2.8

Particle in a three-dimensional box

61

 

Problems

64

Chapter 3

General principles of quantum theory

65

3.1

Linear operators

65

3.2

Eigenfunctions and eigenvalues

67

3.3

Hermitian operators

69

v

vi

Contents

 

3.4

Eigenfunction expansions

75

3.5

Simultaneous eigenfunctions

77

3.6

Hilbert space and Dirac notation

80

3.7

Postulates of quantum mechanics

85

3.8

Parity operator

94

3.9

Hellmann±Feynman theorem

96

3.10

Time dependence of the expectation value

97

3.11

Heisenberg uncertainty principle

99

 

Problems

104

Chapter 4

Harmonic oscillator

106

4.1

Classical treatment

106

4.2

Quantum treatment

109

4.3

Eigenfunctions

114

4.4

Matrix elements

121

4.5

Heisenberg uncertainty relation

125

4.6

Three-dimensional harmonic oscillator

125

 

Problems

128

Chapter 5

Angular momentum

130

5.1

Orbital angular momentum

130

5.2

Generalized angular momentum

132

5.3

Application to orbital angular momentum

138

5.4

The rigid rotor

148

5.5

Magnetic moment

151

 

Problems

155

Chapter 6

The hydrogen atom

156

6.1

Two-particle problem

157

6.2

The hydrogen-like atom

160

6.3

The radial equation

161

6.4

Atomic orbitals

175

6.5

Spectra

187

 

Problems

192

Chapter 7

Spin

194

7.1

Electron spin

194

7.2

Spin angular momentum

196

7.3

Spin one-half

198

7.4

Spin±orbit interaction

201

 

Problems

206

 

Contents

vii

Chapter 8

Systems of identical particles

208

8.1

Permutations of identical particles

208

8.2

Bosons and fermions

217

8.3

Completeness relation

218

8.4

Non-interacting particles

220

8.5

The free-electron gas

226

8.6

Bose±Einstein condensation

229

 

Problems

230

Chapter 9

Approximation methods

232

9.1

Variation method

232

9.2

Linear variation functions

237

9.3

Non-degenerate perturbation theory

239

9.4

Perturbed harmonic oscillator

246

9.5

Degenerate perturbation theory

248

9.6

Ground state of the helium atom

256

 

Problems

260

Chapter 10

Molecular structure

263

10.1

Nuclear structure and motion

263

10.2

Nuclear motion in diatomic molecules

269

 

Problems

279

Appendix A

Mathematical formulas

281

Appendix B

Fourier series and Fourier integral

285

Appendix C

Dirac delta function

292

Appendix D

Hermite polynomials

296

Appendix E

Legendre and associated Legendre polynomials

301

Appendix F

Laguerre and associated Laguerre polynomials

310

Appendix G

Series solutions of differential equations

318

Appendix H

Recurrence relation for hydrogen-atom expectation values

329

Appendix I

Matrices

331

Appendix J

Evaluation of the two-electron interaction integral

341

Selected bibliography

344

Index

 

347

Physical constants

Preface

This book is intended as a text for a ®rst-year physical-chemistry or chemicalphysics graduate course in quantum mechanics. Emphasis is placed on a rigorous mathematical presentation of the principles of quantum mechanics with applications serving as illustrations of the basic theory. The material is normally covered in the ®rst semester of a two-term sequence and is based on the graduate course that I have taught from time to time at the University of Pennsylvania. The book may also be used for independent study and as a reference throughout and beyond the student's academic program.

The ®rst two chapters serve as an introduction to quantum theory. It is assumed that the student has already been exposed to elementary quantum mechanics and to the historical events that led to its development in an undergraduate physical chemistry course or in a course on atomic physics. Accordingly, the historical development of quantum theory is not covered. To serve as a rationale for the postulates of quantum theory, Chapter 1 discusses wave motion and wave packets and then relates particle motion to wave motion. In Chapter 2 the time-dependent and time-independent SchroÈdinger equations are introduced along with a discussion of wave functions for particles in a potential ®eld. Some instructors may wish to omit the ®rst or both of these chapters or to present abbreviated versions.

Chapter 3 is the heart of the book. It presents the postulates of quantum mechanics and the mathematics required for understanding and applying the postulates. This chapter stands on its own and does not require the student to have read Chapters 1 and 2, although some previous knowledge of quantum mechanics from an undergraduate course is highly desirable.

Chapters 4, 5, and 6 discuss basic applications of importance to chemists. In all cases the eigenfunctions and eigenvalues are obtained by means of raising and lowering operators. There are several advantages to using this ladder operator technique over the older procedure of solving a second-order differ-

viii

Preface

ix

ential equation by the series solution method. Ladder operators provide practice for the student in operations that are used in more advanced quantum theory and in advanced statistical mechanics. Moreover, they yield the eigenvalues and eigenfunctions more simply and more directly without the need to introduce generating functions and recursion relations and to consider asymptotic behavior and convergence. Although there is no need to invoke Hermite, Legendre, and Laguerre polynomials when using ladder operators, these functions are nevertheless introduced in the body of the chapters and their properties are discussed in the appendices. For traditionalists, the series-solution method is presented in an appendix.

Chapters 7 and 8 discuss spin and identical particles, respectively, and each chapter introduces an additional postulate. The treatment in Chapter 7 is limited to spin one-half particles, since these are the particles of interest to chemists. Chapter 8 provides the link between quantum mechanics and statistical mechanics. To emphasize that link, the free-electron gas and Bose± Einstein condensation are discussed. Chapter 9 presents two approximation procedures, the variation method and perturbation theory, while Chapter 10 treats molecular structure and nuclear motion.

The ®rst-year graduate course in quantum mechanics is used in many chemistry graduate programs as a vehicle for teaching mathematical analysis. For this reason, this book treats mathematical topics in considerable detail, both in the main text and especially in the appendices. The appendices on Fourier series and the Fourier integral, the Dirac delta function, and matrices discuss these topics independently of their application to quantum mechanics. Moreover, the discussions of Hermite, Legendre, associated Legendre, Laguerre, and associated Laguerre polynomials in Appendices D, E, and F are more comprehensive than the minimum needed for understanding the main text. The intent is to make the book useful as a reference as well as a text.

I should like to thank Corpus Christi College, Cambridge for a Visiting Fellowship, during which part of this book was written. I also thank Simon Capelin, Jo Clegg, Miranda Fyfe, and Peter Waterhouse of the Cambridge University Press for their efforts in producing this book.

Donald D. Fitts

1

The wave function

Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon±the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and super¯uids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new ®eld of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and technology.

Quantum theory was developed during the ®rst half of the twentieth century through the efforts of many scientists. In 1926, E. SchroÈdinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. SchroÈdinger's wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena.

Our presentation of the basic principles of quantum mechanics is contained in the ®rst three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the SchroÈdinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential ®eld. The formal mathematical postulates of quantum theory are presented in Chapter 3.

1