Quantum Mechanics: Foundations and Applications

610 Pages·1986·12.743 MB·English

by  Arno Bohm (auth.)

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 Joseph L. Birman
Institut fUr Angewandte Mathematik Department of Physics
Universităt Heidelberg The City College of the
Im Neuenheimer Feld 5 City University of New York
D-6900 Heidelberg 1 New York, NY 10031
Federal Republic of Germany U.S.A.
Elliott H. Lieb Tullio Regge Walter Thirring
Department of Physics Istituto de Fisica Teorica Institut fUr Theoretische PIiysik
losepIi Henry Laboratories Universita di Torino der Universităt Wien
Princeton University C. so M. d'Azeglio, 46 Boltzmanngasse 5
Princeton, Nl 08540 10125 Torino A-1090 Wien
U.S:A. Italy Austria
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
The first edition ofthis book appeared as: Amo Bohm, Quantum Mechanics. Springer
Veriag, 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 math
ematically 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 under
lying 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 assump
tions 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 improve
ments 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.
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