Texts and Monographs in Physics

w. Beiglbock J. L. Birman E. H. Lieb T. Regge W. Thirring

Series Editors

Texts and Monographs in Physics

R. Bass: Nuclear Reactions with Heavy Ions (1980). A. Bohm: Quantum Mechanics: Foundations and Applications, Second Edition

(1986). O. Bratteli and D. W. Robinson: Operator Algebras and Quantum Statistical

Mechanics. Volume I: C*- and W*-Algebras. Symmetry Groups. Decomposition of States (1979). Volume II: Equilibrium States. Models in Quantum Statistical Mechanics (1981).

K. Chadan and P.C. Sabatier: Inverse Problems in Quantum Scattering Theory (1977).

M. Chaichian and N.F. Nelipa: Introduction to Gauge Field Theories (1984). G. Gallavotti: The Elements of Mechanics (1983). W. Glockle: The Quantum Mechanical Few-Body Problem (1983). W. Greiner, B. Muller, and J. Rafelski: Quantum Electrodynamics of Strong Fields

(1985). J.M. Jauch and F. Rohrlich: The Theory of Photons and Electrons: The Relativistic

Quantum Field Theory of Charged Particles with Spin One-half, Second Expanded Edition (1980).

J. Kessler: Polarized Electrons (1976). Out of print. (Second Edition available as Springer Series in Atoms and Plasmas, Vol. I.)

G. Ludwig: Foundations of Quantum Mechanics I (1983). G. Ludwig: Foundations of Quantum Mechanics II (1985). R.G. Newton: Scattering Theory of Waves and Particles, Second Edition (1982). A. Perelomov: Generalized Coherent States and Their Applications (1986). H. Pilkuhn: Relativistic Particle Physics (1979). R.D. Richtmyer: Principles of Advanced Mathematical Physics. Volume I (1978).

Volume II (1981). W. Rindler: Essential Relativity: Special, General, and Cosmological, Revised

Second Edition (1980). P. Ring and P. Schuck: The Nuclear Many-Body Problem (1980). R.M. Santilli: Foundations of Theoretical Mechanics. Volume I: The Inverse

Problem in Newtonian Mechanics (1978). Volume II: Birkhoffian Generalization of Hamiltonian Mechanics (1983).

M.D. Scadron: Advanced Quantum Theory and Its Applications Through Feynman Diagrams (1979).

N. Straumann: General Rl!lativity and Relativistic Astrophysics (1984). C. Truesdell and S. Bharatha: The Concepts and Logic of Classical

Thermodynamics as a Theory of Heat Engines: Rigourously Constructed upon the Foundation Laid by S. Carnot and F. Reech (1977).

F.J. Ynduniin: Quantum Chromodynamics: An Introduction to the Theory of Quarks and Gluons (1983).

Arno Bohm

QuantUITl Mechanics: F oundations and Applications

Second Edition, Revised and Enlarged Prepared with M. Loewe

With 94 Illustrations

6 Springer Science+Business Media, LLC

Arno Bohm Department of Physics Center for Particle Theory The University of Texas at Austin Austin, TX 78712 U.S.A.

Editors

Wolf Beiglbock Institut fUr Angewandte Mathematik Universităt Heidelberg

Joseph L. Birman Department of Physics The City College of the

Im Neuenheimer Feld 5 D-6900 Heidelberg 1

City University of New York New York, NY 10031

Federal Republic of Germany

Elliott H. Lieb Department of Physics losepIi Henry Laboratories Princeton University Princeton, Nl 08540 U.S:A.

U.S.A.

Tullio Regge Istituto de Fisica Teorica Universita di Torino C. so M. d'Azeglio, 46 10125 Torino Italy

Library of Congress Cataloging in Publieation Data Bohm, Amo, 1936~

Quantum meehanies. (Texts and monographs in physies) Bibliography: p. Includes index. 1. Quantum theory. 1. Title. II. Series

QCI74.12.B63 1986 530.1'2 85-4710

Walter Thirring Institut fUr Theoretische PIiysik

der Universităt Wien Boltzmanngasse 5 A-1090 Wien Austria

The first edition ofthis book appeared as: Amo Bohm, Quantum Mechanics. Springer­Veriag, New York, Heidelberg, Berlin, 1979.

© 1979, 1986 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Ine. in 1986 Softcover reprint ofthe hardcover 2nd edition 1986

AlI rights reserved. No part of this book may be translated or reprodueed in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, V.S.A.

Typeset by Composition House Ltd., Salisbury, England.

9 8 7 6 5 4 3 2

ISBN 978-3-540-13985-0 ISBN 978-3-662-01168-3 (eBook) DOI 10.1007/978-3-662-01168-3

To my students, colleagues, and friends without whose help this second edition

would not have been possible.

Preface

The first edition of this book was written as a text and has been used many times in a one-year graduate quantum mechanics course. One of the reviewers has made me aware that the book can also serve as, " ... in principle, a handbook of nonrelativistic quantum mechanics." In the second edition we have therefore added material to enhance its usefulness as a handbook. But it can still be used as a text if certain chapters and sections are ignored. We have also revised the original presentation, in many places at the suggestion of students or colleagues. As a consequence, the contents of the book now exceed the material that can be covered in a one-year quantum mechanics course on the graduate level. But one can easily select the material for a one-year course omitting-according to one's preference-one or several of the following sets of sections: {1.7, XXI}, {X, XI} or just {XI}, {II.7, XIII}, {XIV.5, XV}, {XIX, XX}. Also the material of Sections 1.5-1.8 is not needed to start with the physics in Chapter II. Chapters XI, XIII, XIX, and XX are probably the easiest to dispense with and I was contemplating the deletion of some of them, but each chapter found enthusiastic supporters among the readers who advised against it.

Chapter I-augmented with some applications from later chapters-can also be used as a separate introductory text on the mathematics of quantum mechanics.

The book is self-contained and does not require any prior knowledge of quantum mechanics, but it is a difficult book, because it is so concise. It offers a huge amount of material, more than one can find in texts with twice the number of pages. Consequently, some familiarity with the subject would be

vii

viii Preface

very helpful. Prerequisites are a knowledge of calculus, vector algebra, and analysis. Most physical examples are taken from the fields of atomic and molecular physics, as it is these fields that are best known to students at the stage when they learn quantum mechanics.

Texts on a subject established a half-century before are often written using the material and presentation established by the first generation of books on that subject. New applications, deeper insights, and unifying formulations that subsequently develop are easily overlooked. That has not been done in this text. I have presented a unified theoretical formulation which was made possible by later developments, and have included examples from more recent papers.

The changes incorporated in the second edition provide an easier access to the material, but leave the general idea unchanged. It is therefore fitting to quote from the preface of the first edition:

... in contrast to what one finds in the standard books, quantum mechanics is more than the overemphasized wave-particle dualism presented in the familiar mathematics of differential equations. "This latter dualism is only part of a more general pluralism" (Wigner) because, besides momentum and position, there is a plurality of other observables not commuting with position and momentum. As there is no principle that brings into prominence the position and momentum operators, a general formalism of quantum mechanics, in which every observable receives the emphasis it deserves for the particular problem being considered, is not only preferable but often much more practical .... It is this general form of quantum theory that is presented here.

I have attempted to present the whole range from the fundamental assumptions to the experimental numbers. To do this in the limited space available required compromises. My choice ... was mainly influenced by what I thought was needed for modern physics and by what I found, or did not find, in the standard textbooks. Detailed discussions of the Schrodinger differential equation for the hydrogen atom and other potentials can be found in many good books.! On the other hand, the descriptions of the vibrational and rotational spectra of molecules are hardly treated in any textbooks of quantum mechanics, though they serve as simple examples for the important procedure of quantum-mechanical model building .... So I have treated the former rather briefly and devoted considerable space to the latter.

Groups have not been explicitly made use of in this book. However, the reader familiar with this subject will see that group theory is behind most of the statements that have been cast here in terms of algebras of observables.

1 The subject also is usually adequately treated in undergraduate courses.

Preface ix

This is a physics book, and though mathematics has been used extensively, I have not endeavored to make the presentation math­ematically rigorous .... Except in the mathematical inserts, which are given in openface brackets [M: ], the reader will not even be made aware of these mathematical details.

The mathematical inserts are of two kinds. The first kind provides the mathematics needed, and the second kind indicates the under­lying mathematical justification ....

Quantum mechanics starts with Chapter II, where the most essential basic assumptions (axioms) of quantum mechanics are made plausible from the example of the harmonic oscillator as realized by the diatomic molecule. Further basic assumptions are introduced in later chapters when the scope of the theory is extended. These basic assumptions ("postulates") are not to be understood as mathematical axioms from which everything can be derived without using further judgment and creativity. An axiomatic approach of this kind does not appear to be possible in physics. The basic assump­tions are to be considered as a concise way of formulating the quintessence of many experimental facts.

The book consists of two clearly distinct parts, Chapters II-XI and Chapters XIV-XXI, with two intermediate chapters, Chapters XII and XIII. The first part is more elementary in presentation, though more fundamental in subject matter .... The second part, which starts with Chapter XIV, treats scattering and decaying systems. The presentation there is more advanced.

Chapter XIV gives a derivation of the cross section under very general conditions .... Two different points of view-one in which the Hamiltonian time development is assumed to exist, and the other making use of the S-matrix-are treated in a parallel fashion. The required analyticity of the S-matrix is deduced from causality. One of the main features of the presentation is to treat discrete and continuous spectra from the same point of view. For this the rigged Hilbert space is needed, which provides not only a mathematical simplification but also a description which is closer to physics.

Major changes to the book have been made in Chapters I, XIII, and XXI, which were almost totally rewritten. Chapter XXI discusses the new notion of Gamow vectors for the description of decaying states. They were created when the first edition was written in order to achieve the desired unity of description of all of quantum mechanics. Chapter I had to be expanded to provide the mathematical background for Chapter XXI. To start a physics book with a mathematical introduction may create an incorrect impression. I therefore want to emphasize that the book contains many more experimental numbers than mathematical theorems. Extensive revisions have also been made in Chapters II, IV, XIV, XVI, XVII, and XVIII; and many improve­ments were made in Chapters III, V, VIII, and IX. The appendix to Section

x Preface

V.3 has been rewritten to provide a simple but typical example for the construction of noncom pact group representations. Not all chapters could be revised because of time limitations. Chapters VII, X, XI, and XIX have been scrutinized only a little and Chapters VI, XII, XV, and XX remain essentially as they were in the first edition.

Acknowledgments

For the second edition, as for the first edition, I am indebted to many for their help, encouragement, and advice. Chapter XIII was rewritten with K. Kraus, who together with A. Peres also suggested improvements to Chapter II. On the material of Chapter XXI, I received advice from L. Khalfin and M. Gadella. The new version of Chapter I grew out of a joint project with G. B. Mainland. The revisions of the first part of the book were made together with M. Loewe. For the revisions of the second part of the book I was assisted by J. Morse. P. Busch proofread Section XIII.l. I received many letters pointing out misprints and inadequacies, suggesting improvements, and encouraging me through the tedious task of preparing a new edition. I would like to thank R. Scalettar, A. Y. Klimik, L. Fonda, and T. Mertelmeier for pointing out errors in the first edition. The numerous misprints could not have been purged without the help of students in my classes. Support from D.O.E. and the Alexander von Humboldt Foundation is gratefully acknowledged. I am particularly grateful to M. Loewe who proofread the entire book and made many improvements. If the second edition is better than the first, it is mainly due to him.

xi

Contents

CHAPTER I Mathematical Preliminaries 1.1 The Mathematical Language of Quantum Mechanics 1 1.2 Linear Spaces, Scalar Product 2 1.3 Linear Operators 5 1.4 Basis Systems and Eigenvector Decomposition 8 1.5 Realizations of Operators and of Linear Spaces 18 1.6 Hermite Polynomials as an Example of Orthonormal Basis Functions 28

Appendix to Section 1.6 31 1.7 Continuous Functionals 33 1.8 How the Mathematical Quantities Will Be Used 39

Problems 39

CHAPTER II Foundations of Quantum Mechanics-The Harmonic Oscillator 43

II. I Introduction 43 11.2 The First Postulate of Quantum Mechanics 44 11.3 Algebra of the Harmonic Oscillator 50 11.4 The Relation Between Experimental Data and Quantum-Mechanical

Observables 54 11.5 The Basic Assumptions Applied to the Harmonic Oscillator, and

Some Historical Remarks 74 11.6 Some General Consequences of the Basic Assumptions of Quantum

Mechanics 81 11.7 Eigenvectors of Position and Momentum Operators; the Wave

Functions of the Harmonic Oscillator 84

X1ll

XIV Contents

II.8 Postulates II and III for Observables with Continuous Spectra II.9 Position and Momentum Measurements-Particles and Waves

Problems

CHAPTER III

94 101 112

Energy Spectra of Some Molecules 117 III.1 Transitions Between Energy Levels of Vibrating Molecules-

The Limitations of the Oscillator Model 117 III.2 The Rigid Rotator 128 III.3 The Algebra of Angular Momentum 132 III.4 Rotation Spectra 138 III.5 Combination of Quantum Physical Systems-The Vibrating Rotator 146

Problems 155

CHAPTER IV

Complete Systems of Commuting Observables 159

CHAPTER V

Addition of Angular Momenta-The Wigner-Eckart Theorem 164 V .1 Introduction-The Elementary Rotator !64 V.2 Combination of Elementary Rotators 165 V.3 Tensor Operators and the Wigner-Eckart Theorem 176

Appendix to Section V.3 181 V.4 Parity 192

Problem 204

CHAPTER VI

Hydrogen Atom-The Quantum-Mechanical Kepler Problem 205 VI.I Introduction 205 VI.2 Classical Kepler Problem 206 VI.3 Quantum-Mechanical Kepler Problem 208 VI.4 Properties of the Algebra of Angular Momentum and the Lenz Vector 213 VI.5 The Hydrogen Spectrum 215

Problem 222

CHAPTER VII

Alkali Atoms and the Schr6dinger Equation of One-Electron Atoms 223 VII.! The Alkali Hamiltonian and Perturbation Theory 223 VII.2 Calculation of the Matrix Elements of the Operator Q -, 227 VII.3 Wave Functions and Schrodinger Equation of the Hydrogen Atom

and the Alkali Atoms 234 Problem 241

CHAPTER VIII

Perturbation Theory VIlLI Perturbation of the Discrete Spectrum VIII.2 Perturbation of the Continuous Spectrum-

The Lippman-Schwinger Equation Problems

242 242

248 251

Contents xv

CHAPTER IX Electron Spin IX.l Introduction IX.2 The Fine Structure-Qualitative Considerations IX.3 Fine-Structure Interaction IX.4 Fine Structure of Atomic Spectra IX.5 Selection Rules IX.6 Remarks on the State of an Electron in Atoms

Problems

CHAPTER X

253 253 255 261 268 270 271 272

Indistinguishable Particles 274 X.l Introduction 274

Problem 281

CHAPTER XI Two-Electron Systems-The Helium Atom 282 XL! The Two Antisymmetric Subspaces of the Helium Atom 282 XL2 Discrete Energy Levels of Helium 287 XI.3 Selection Rules and Singlet-Triplet Mixing for the Helium Atom 297 XL4 Doubly Excited States of Helium 303

Problems 309

CHAPTER XII

Time Evolution 310 XII.l Time Evolution 310 XII.A Mathematical Appendix: Definitions and Properties of Operators

that Depend upon a Parameter 324 Problems 326

CHAPTER XIII Some Fundamental Properties of Quantum Mechanics 328

XIII. I Change of the State by the Dynamical Law and by the Measuring Process-The Stern-Gerlach Experiment 328 Appendix to Section XIII. 1 340

XIII.2 Spin Correlations in a Singlet State 342 XIII.3 Bell's Inequalities, Hidden Variables, and the Einstein-Podolsky-

Rosen Paradox 347 Problems 354

CHAPTER XIV Transitions in Quantum Physical Systems-Cross Section 356 XIV.! Introduction 356 XIV.2 Transition Probabilities and Transition Rates 358 XIV.3 Cross Sections 362 XIV.4 The Relation of Cross Sections to the Fundamental Physical

Observables 365 XIV.5 Derivation of Cross-Section Formulas for the Scattering of

a Beam off a Fixed Target Problems

368 384

xvi Contents

CHAPTER XV

Formal Scattering Theory and Other Theoretical Considerations 387

XV. I The Lippman-Schwinger Equation 387 XV.2 In-States and Out-States 391 XV.3 The S-Operator and the Melller Wave Operators 399 XV.A Appendix 407

CHAPTER XVI

Elastic and Inelastic Scattering for Spherically Symmetric Interactions XVI.I Partial-Wave Expansion XVI.2 Unitarity and Phase Shifts XVI.3 Argand Diagrams

Problems

CHAPTER XVII

Free and Exact Radial Wave Functions

XVII.I Introduction XVII.2 The Radial Wave Equation XVII.3 The Free Radial Wave Function XVII.4 XVII.5 XVII.6

XVII. A

The Exact Radial Wave Function Poles and Bound States Survey of Some General Properties of Scattering Amplitudes and Phase Shifts Mathematical Appendix on Analytic Functions Problems

CHAPTER XVlIl

409

409 417 422 424

425

425 426 430 432 439

441 444 450

Resonance Phenomena 452

XVIII. I Introduction 452 XVIII.2 Time Delay and Phase Shifts 457 XVIII. 3 Causality Conditions 464 XVIII.4 Causality and Analyticity 467 XVIII.5 Brief Description of the Analyticity Properties of the S-Matrix 471 XVIII.6 Resonance Scattering-Breit-Wigner Formula for Elastic Scattering 476 XVIII. 7 The Physical Effect of a Virtual State 487 XVIII. 8 Argand Diagrams for Elastic Resonances and Phase-Shift Analysis 489 XVIII.9 Comparison with the Observed Cross Section: The Effect of

Background and Finite Energy Resolution 493 Problems 503

CHAPTER XIX

Time Reversal XIX.l Space-Inversion Invariance and the Properties of the S-Matrix XIX.2 Time Reversal

Appendix to Section XIX.2 XIX.3 Time-Reversal Invariance and the Properties of the S-Matrix

Problems

505

505 507 511 512 516

Contents XVll

CHAPTER XX

Resonances in Multichannel Systems 517 XX.! Introduction 517 XX.2 Single and Double Resonances 518 XX.3 Argand Diagrams for Inelastic Resonances 532

CHAPTER XXI

The Decay of Unstable Physical Systems 537 XXI.I Introduction 537 XXI.2 Lifetime and Decay Rate 539 XXI.3 The Description of a Decaying State and the Exponential Decay Law 542 XXI.4 Gamow Vectors and Their Association to the Resonance Poles of the

S-Matrix 549 XXl.5 The Golden Rule 563 XXI.6 Partial Decay Rates 567

Problems 569

Epilogue 571

Bibliography 574

Index 579