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Quantum Mechanics - Schiff

QUANTUM MECHANICS

INTERNATIONAL SERIESLeonardI.

IN

PURE AND APPLIED PHYSICS

Schiff,

Consulting Editor

ADLER, BAZIN, and SCHIFFER Introduction to General Relativity ALUS and HERLIN Thermodynamics and Statistical Mechanics BECKER Introduction to Theoretical Mechanics BJORKEN and DRELL Relativistic Quantum Fields BJORKEN and DRELL Relativistic Quantum Mechanics CHODOROW and SUSSKIND Fundamentals of Microwave Electronics

CLARKCOLLIN

Applied X-raysField

Theory

of

Guided WavesIntegrals

EVANS The Atomic Nucleus FEYNMAN and HI BBS Quantum Mechanics and Path GINZTON Microwave Measurements HALL Introduction to Electron Microscopy HARDY and PERRIN The Principles of Optics

HARNWELL Electricity and Electromagnetism HARNWELL and LIVINGOOD Experimental AtomicHENLEY and THIRRING

Physics

HOUSTON JAMMER KENNARDLEIGHTON

Elementary Quantum Field Theory Principles of Mathematical Physics The Conceptual Development of Quantum Mechanics Kinetic Theory of Gases

Principles of Modern Physics LINDSAY Mechanical Radiation LIVINGSTON and BLEWETT Particle Accelerators MIDDLETON An Introduction to Statistical Communication Theory MORSE Vibration and Sound MORSE and FESHBACH Methods of Theoretical Physics

MORSE and INGARD Theoretical Acoustics MUSKAT Physical Principles of Oil Production NEWTON Scattering Theory of Waves and ParticlesPRESENT Kinetic Theory of Gases READ Dislocations in Crystals RICHTMYER, KENNARD, and LAURITSEN SCHIFF Quantum MechanicsIntroduction to

Modern Physics

SCHWARTZSEITZ

The Modern Theory

SLATER SLATER SLATER SLATER SLATER SLATER SLATER

Introduction to Special Relativity of Solids Introduction to Chemical Physicsofof of

Quantum Theory Quantum Theory Quantum Theory

Atomic Structure, Atomic Structure, Matterin

Vol.

I

Vol.

II

Electronic Structure of Molecules:

Symmetry and Energy BandsSolids, Vol. 2

Quantum Theory of Molecules and Solids, Vol. 1 Crystals: Quantum Theory of Molecules andof Molecules

Insulators, Semiconductors,Solids, Vol. 3

and Metals: Quantum Theory

and

Mechanics SMYTH E Static and Dynamic Electricity STRATTON Electromagnetic Theory TINKHAM Group Theory and Quantum Mechanics TOWNES and SCHAWLOW Microwave Spectroscopy

SLATER and FRANK SLATER and FRANK

Introduction to Theoretical Physics

WANGWHITEThe

Solid-state Electronics

Introduction to Atomic Spectra

late F. K. Richtmyer was Consulting Editor of the series from its inception in 1929 to his death in 1939. Lee A. Du Bridge was Consulting Editor from 1939 to 1946; and G. P. Harnwell from 1947 to 1954.

QUANTUMMECHANICSTHIRD EDITION

LEONARD

I.

SCHIFF

Professor of Physics

Stanford University

INTERNATIONAL STUDENT EDITION

McGRAW-HILL BOOK COMPANYNewYorkSt.

Louis

San Francisco

London

Mexico

Panama

Sydney

Toronto

KOGAKUSHA COMPANY, LTD.Tokyo

QUANTUM MECHANICSINTERNATIONAL STUDENT EDITIONExclusive rights by Kogakusha Co., Ltd., for manufacture and export from Japan. This book cannot be re-exported from the country to which it is consigned by Kogakusha Co., Ltd., or by McGraw-Hill Book Company or any of its subsidiaries.

vCopyright 1955, 1968 by McGraw-Hill, Inc. All Rights Reserved. Copyright 1949 by McGraw-Hill, Inc. All Rights Reserved.

Noin

part of this publication

may

be reproduced,electronic,

stored in a retrieval system, or transmitted,

any form or by any means,

mechanical, photocopying, recording, or otherwise, without the prior written permissionof the publisher.

Library of Congress Catalog Card Number 68-25665

TOSIIO PRINTING CO., LTD., TOKYO,

JAPAN

TO FRANCES

PREFACE

This volume has a threefold purpose; to explain the physical concepts of quantum mechanics, to describe the mathematical formalism, and to The present illustrative examples of both the ideas and the methods. book is intended to serve as a text at the graduate level and also as a It is assumed that the reader is reasonably familiar with reference book. classical mechanics, electromagnetic theory, atomic structure, and differential equations; prior acquaintance with matrices or group theory is not necessary. In addition, he should have had some contact with complex variables (for Chap. 9) and the special theory of relativity (for Chap. 13). The author believes that the analytical methods employed in the book will satisfy most theoretical physicists even though no attempt is made to achieve mathematical rigor. For example, there is little or no discussion of the justification for the interchange of sum, derivative, and On the other hand, integral operations or for the use of the 5 function. the physical reasons for the nature of the results obtained are investigated

wherever possible. Problems are given at the end of each chapter. They are often used Original theoretical to illustrate or amplify points discussed in the text. papers are referred to throughout the book; the list is representative Experimental results are, for the most part, rather than exhaustive. quoted without reference, since the large amount of documentation required for an adequate survey seems out of place in a book on theoretical Several other books and review articles on quantum mechanics physics.

and

related subjects are referred to for

more detailed discussions ofif

particular topics.

The scope

of this

volume

is

best outlined

the book

is

divided intoix

;

x

PREFACE

The first three chapters constitute an introduction to quantum mechanics, m which the physical concepts are discussed and the Schrodinger wave formalism is established. The next nine chapters present exact solutions of the wave equation for both bound-state andthree parts.collision problems, the

Heisenberg matrix formalism and transformation

theory, symmetry, approximation methods, the scattering matrix, particleidentity,

radiation theory, and

some applications to atomic systems.

Since Chaps. 5 to 12 include most of the material given in a first-year

graduate course, it seems desirable to include a semiclassical treatment of electromagnetic radiation (Chap. 11) even though some of the results are obtained again in Chap. 14. The last two chapters are an introduction to relativistic particle theory and to quantized fields. The first edition of this book was completed 20 years ago, and Thus the present relatively few changes were made in the second edition.revisionis of necessity extensive; at the same time it is intended to retain the comprehensiveness of the original volume without a substantial increase in length. The principal additions are a section on complex

and the reciprocity and optical theorems (Sec. 20); a much account of matrices and transformation theory (Chap. 6); a new chapter that discusses geometrical and dynamical symmetries and includes a fairly detailed account of angular momentum (Chap. 7); a considerably expanded treatment of approximation methods for boundpotentialsfuller

problems, including the scattering matrix and its and dispersion relations (Chaps. 8 and 9) and a new section on the density operator and matrix (Sec. 42). The principal topics dropped from the second edition are the variational treatment of scattering, the theory of the Cerenkov effect, and the quantizastate

and

collision

applications, analytic properties,

tion of the Dirac equation; also, the last

are

somewhat condensed and combined

volume.

Some changes

in

two sections of the second edition into one (Sec. 57) in the present notation have been made to conform to cur-

rent usage, and a table of the numerical values of

some physical

quantities

has been added inside the back cover. The author wishes again to record his indebtedness to the late Prof. J. R. Oppenheimer and to Prof. Robert Serber in connection with He is also grateful to the preparation of the first edition of this book. several of those who have studied and taught from the earlier revision for In particular, Prof. E. H. Wichmann their many helpful suggestions. prepared a thorough review of the second edition that contributed substantially to the present volume. The author also thanks the many

who studied from the various drafts of the third edition for their comments, and especially Prof. J. D. Walecka for his constructive critistudents

cism of particular sections.

Leonard

I.

Schiff

CONTENTS

Preface

ix

Chapter1

1

The Physical Basis

of

Quantum MechanicsSummaryexperiments

1

2

Experimental Background Inadequacy of classical physics. and inferences, The Old Quantum Theory Bohr-Sommerfeld quantizationceptualdifficulties.

2of principal

4rules.

Practical difficulties.

Con7

Quantum- mechanical viewpoint.principle.

3

4

Uncertainty and Complementarity Uncertainty principle. Complementarity on experiment. Discussion of MeasurementLocalization

Limitations9

experiment.diffraction

Momentumexperiment.

determination experiment.Discussionof

Analysis5

of

diffraction

experiment.

Wave PacketsSpace packets.

in

Space and Time Time packets. Wave formalism.

14