The Schr¶dinger Equation
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Managing Editor:
M. HAZEWINKEL Centre for Mathematics and Computer Science,
Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV. MGU, Moscow, U.S.S.R. Yu. I. MANIN, Steklov
Institute of Mathematics, Moscow, U.S.S.R. N. N. MOISEEV, Computing
Centre, Academy ofSciences, Moscow, U.S.S.R. S. P. NOVIKOV, Landau
Institute ofTheoretical Physics, Moscow, U.S.S.R. M. C. POLYVANOV.
Steklov Institute of Mathematics, Moscow, U.S.S.R. Yu. A. ROZANOV,
Steklol' Institute ofMathematics, Moscow, U.S.S.R.
Volume 66
and
M.A. Shubin Center for Optimization and Mathematical Modelling,
Institute of New Technologies, Moscow, U.S.S.R.
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Berezin. F. A. (Feliks Aleksanarovich) The Schroainger equation I
by F.A. Berezin ana M.A. Shubin with
the assistance of G.L. Litvinov and O.A. Leites. p. cm. --
(Mathematics ana its applications (Soviet series)
v. 66) Incluaes bibliographical references and index. ISBN
978-94-010-5391-4 ISBN 978-94-011-3154-4 (eBook) DOI
10.1007/978-94-011-3154-4 1. Schrodinger equation. 1. Shubin. M. A.
(Mikhail
Aleksanarovichl. 1944- II. Title. III. Series: Mathematics and its
applications (Kluwer Academie Publishers). Soviet series ; 66.
QCI74.26.W28B45 1991 530. 1 '24--dc20 91-11946
ISBN 978-94-010-5391-4
Printed on acid-free paper
This English edition is a revised, expanded version of the original
Soviet publication.
This is the translation of the work YPABHEHI1E lllPE.nI1HfEPA
Published by the Moscow State University, Moscow, © 1983.
Translated from the Russian by Yu. Rajabov, D. A. Leites and N. A.
Sakharova
AII Rights Reserved © 1991 Springer Science+Business Media
Dordrecht Originally published by Kluwer Academic Publishers in
1991 Softcover reprint ofthe hardcover Ist edition 1991 No part of
the material protected by this copyright notice may be reproduced
or utilized in any form or by any means, electronic or mechanical,
inc\uding photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright
owner.
SERIES EDITOR'S PREFACE
4Et moi, ..., si j'avait su comment en revenir, je n'y serais point
alle.'
Jules Verne
The series is divergent; therefore we may be able to do something
with it.
O. Heaviside
One service mathematics has rendered the human race. It has put
common sense back where it belongs, on the topmost shelf next to
the dusty canister labelled 'discarded non· sense'.
Eric T. Bell
Mathematics is a tool for thought. A highly necessary tool in a
world where both feedback and non linearities abound. Similarly,
all kinds of parts of mathematics serve as tools for other parts
and for other sciences. Applying a simple rewriting rule to the
quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics ...'; 'One
service logic has rendered com puter science ...'; 'One service
category theory has rendered mathematics ...'. All arguably true.
And all statements obtainable this way form part of the raison
d'€tre of this series. This series, Mathematics and Its
ApplicatiOns, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At
the time I wrote
"Growing specialization and diversification have brought a host of
monographs and textbooks on increasingly specialized topics.
However, the 'tree' of knowledge of mathematics and related fields
does not grow only by putting forth new branches. It also happens,
quite often in fact, that branches which were thought to be
completely disparate are suddenly seen to be related. Further, the
kind and level of sophistication of mathematics applied in various
sciences has changed drastically in recent years: measure theory is
used (non-trivially) in regional and theoretical economics;
algebraic geometry interacts with physics; the Minkowsky lemma,
coding theory and the structure of water meet one another in
packing and covering theory; quantum fields, crystal defects and
mathematical programming profit from homotopy theory; lie algebras
are relevant to filtering; and prediction and electrical
engineering can use Stein spaces. And in addition to this there are
such new emerging subdisciplines as 'experimental mathematics',
'CFD', 'completely integrable systems', 'chaos, synergetics and
large-scale order', which are almost impossible to fit into the
existing classification schemes. They draw upon widely different
sections of mathematics."
By and large, all this still applies today. It is still true that
at first sight mathematics seems rather fragmented and that to
find, see, and exploit the deeper underlying interrelations more
effort is needed and so are books that can help mathematicians and
scientists do so. Accordingly MIA will continue to try to make such
books available.
If anything, the description I gave in 1977 is now an
understatement. To the examples of interaction areas one should add
string theory where Riemann surfaces, algebraic geometry, modu lar
functions, knots, quantum field theory, Kac-Moody algebras,
monstrous moonshine (and more) all come together. And to the
examples of things which can be usefully applied let me add the
topic 'finite geometry'; a combination of words which sounds like
it might not even exist, let alone be applicable. And yet it is
being applied: to statistics via designs, to radar/sonar detection
arrays (via finite projective planes), and to bus connections of
VLSI chips (via difference sets). There seems to be no part of
(so-called pure) mathematics that is not in immediate danger of
being applied. And, accordingly, the applied mathematician needs to
be aware of much more. Besides analysis and numerics, the
traditional workhorses, he may need all kinds of combinatorics,
algebra, probability, and so on. In addition, the applied scientist
needs to cope increasingly with the nonlinear world and the
vi SERIES EDITOR'S PREFACE
extra mathematical sophistication that this requires. For that is
where the rewards are. Linear models are honest and a bit sad and
depressing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic
outputs (or vice versa). To appreci ate what I am hinting at: if
electronics were linear we would have no fun with transistors and
com puters; we would have no TV; in fact you would not be reading
these lines. There is also no safety in ignoring such outlandish
things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All
three have applications in both electrical engineering and physics.
Once, complex numbers were equally outlandish, but they fre
quently proved the shortest path between 'real' results. Similarly,
the first two topics named have already provided a number of
'wormhole' paths. There is no telling where all this is leading
fortunately. Thus the original scope of the series, which for
various (sound) reasons now comprises five sub
series: white (Japan), yellow (China), red (USSR), blue (Eastern
Europe), and green (everything else), still applies. It has been
enlarged a bit to include books treating of the tools from one
subdis cipline which are used in others. Thus the series still
aims at books dealing with:
- a central concept which plays an important role in several
different mathematical and/or scientific specialization areas; -
new applications of the results and ideas from one area of
scientific endeavour into another; .. influences which the results,
problems and concepts of one field of enquiry have, and have had,
on the development of another.
That the SchrOdinger equation is central to all of quantum
mechanics is nothing new. That it is mathematically a rich and
complicated equation also not. In fact it better be, as, in a way,
it is the equation of everything microphysical. That makes writing
about it, and about quantum mechanics, difficult, and virtually all
books on the topic sacrifice mathematical rigour and, especially,
precise statements. This is distressing to mathematicians and puts
them off; it makes it difficult for mathematicians to feel at home
in the quantum world. This book by the late FA Berezin, the origi
nator of a great many ideas in supersymmetry and second
quantization and top analyst M.A. Shu bin is an exception.
Wherever possible the utmost mathematical precision is used. I have
to say 'wherever possible' because there still are parts where more
research is needed to make things rigorous and mathematically
satisfactory. A foremost example of that is the theory of one of
the central tools, the Feynman path integral (to which a large
chapter is devoted). Here, it is nice to note that there has been a
great deal of progress recently based on T. Hida's white noise
analysis (infinite-dimensional stochastic calculus). A great deal
of sophisticated mathematics gets involved when one takes the
Scbrodinger equation seriously, and, as the book starts at the
graduate student level and ends with the most modem developments
such as supersymmetry and supermanifolds, it has become a rather
large volume. It will take time to study it completely but for
those who desire to feel comfortable in the quantum world that time
will be an optimal investment.
The shortest path between two truths in the
real domain passes through the complex
domain.
J. Hadamard
La physique ne nous donne pas seulement l'occasion de r.:soudre des
problemes ... eIle
nous fOO t pressentir la solution.
H. Poincare
them; the only books I have in my library
are books that other folk have lent me.
Anatole FlllJ1ce
The function of an expert is not to be more
right than other people, but to be wrong for
more sophisticated reasons.
Introduction . . . . . . . . . . . . . . 1
1.2. Some Corollaries of the Basic Postulates 8
1.3. Time Differentiation of Observables . . 14
1.4. Quantization . . . . . . . . . . . . 17
Physical Quantities . . . . . . . . . . . . 23
1.7. Particles with Spin 30
1.8. Harmonic Oscillator . 33
1.9. Identical Particles . . 40
1.10. Second Quantization . 44
2.1. Self-Adjointness . . . . . . . . . . . . . . . . . . .
50
2.2. An Estimate of the Growth of Generalized Eigenfunctions
55
2.3. The Schrodinger Operator with Increasing Potential . . .
57
1. Discreteness of spectrum (57). 2. Comparison theorems and
the
behaviour of eigenfunctions as x - 00. (59). 3. Theorems on zeros
of
eigenfunctions (64).
2.4. On the Asymptotic Behaviour of Solutions of Certain
Second-Order
Differential Equations as x - 00 .. . . . . . . . . . . . ..
69
1. The case of integrable potential (70). 2. Liouville's
transformation
and operators with non-integrable potential (81).
Vill CONTENTS
2.5. On Discrete Energy Levels of an Operator with
Semi-Bounded
Potential . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
1 The operator in a half-axis with Dirichlet's boundary condition
(87).
2. The case of an operator on the half-axis with the Neumann
bound-
ary condition (94). 3. The case of an operator on the whole axis
(97).
2.6. Eigenfunction Expansion for Operators with Decaying Potentials
99
1. Preliminary remarks (99). 2. Formulation of the main
theorem
(102). 3. Two proofs of Theorem 6.1. (102). 4. One-dimensional
oper-
ator obtained from the radially symmetric three-dimensional
operator
(116). 5. The case of an operator on the whole axis (122).
2.7. The Inverse Problem of Scattering Theory . . . . . . . . . . .
126
1. Inverse problem on the half-axis (127). 2. Inverse problem on
the
whole axis (131).
2.8. Operator with Periodic Potential . . . . . . . . . . . . . . .
134
1. Bloch functions and the band structure of the spectrum
(134).
2. Expansion into Bloch eigenfunctions (141). 3. The density of
states
(145).
3.1. Self-Adjointness. . . . . . . . . . . . . . 150
3.2. An Estimate of the Generalized Eigenfunctions . . . . . .
160
3.3. Discrete Spectrum and Decay of Eigenfunctions . . . . . .
164
1. Discreteness of spectrum (165). 2. Decay of eigenfunctions
(167). 3. Non-degeneracy of the ground state and positiveness of
the first
eigenfunction (177). 4. On the zeros of eigenfunctions (180).
3.4. The Schrodinger Operator with Decaying Potential: Essential
Spec
trum and Eigenvalues . . . . . . . . . . . . . . . . . . . .
181
1. Essential spectrum (182). 2. Separation of variables in the case
of
spherically symmetric potential and the Laplace-Beltrami operator
on
a sphere (183). 3. Estimation of the number of negative
eigenvalues
(189).4. Absence of positive eigenvalues (191).
3.5. The Schrodinger Operator with Periodic Potential . . . . . . .
. 198
1. Lattices (198). 2. Bloch functions (200). 3. Expansion in Bloch
func
tions (204). 4. Band functions and the band structure of the
spectrum
(208). 5. Theorem on eigenfunction expansion (213). 6.
Non-triviality
of band functions and the absence of a point spectrum (216). 7.
Den-
sity of states (220).
4.1. The Wave Operators and the Scattering Operator . . . . . . . .
223
1. The basic definitions and the statement of the problem
(223).
2. Physical interpretation (225). 3. Properties of the wave
operators
(226). 4. The invariance principle and the abstract conditions for
the
existence and completeness of the wave operators (230).
4.2. Existence and Completeness of the Wave Operators . . . . . . .
233
1. The abstract scheme of Enss (233). 2. The case of the
Schrodinger
operator (242). 3. The scattering matrix (249). 4.
One-dimensional
case (252). 5. Spherically symmetric case (256).
4.3. The Lippman-Schwinger Equations and the Asymptotics of
Eigen
functions . . . . . . . . . . . . . . . . . . . . . . . . . .
259
1. A derivation of the Lippman-Schwinger equations (259). 2.
Another
derivation of the Lippman-Schwinger equations (262). 3. An
outline
of the proof of the completeness of wave operators by the
station-
ary method (265). 4. Discussion on the Lippman-Schwinger
equation
(271). 5. Asymptotics of eigenfunctions (279).
CHAPTER 5. Symbols of Operators and Feynman Path Integrals
282
5.1. Symbols of Operators and Quantization: qp- and pq-Symbols
and
Weyl Symbols 282
1. The general concept of symbol and its connection with
quantization
(282).2. The qp- and pq-symbols (285). 3. Symmetric or Weyl
symbols
(294). 4. Weyl symbols and linear canonical transformations
(300).
5. Weyl symbols and reflections (302).
5.2. Wick and Anti-Wick Symbols. Covariant and Contravariant
Symbols 304
1. Annihilation and creation operators. Fock space (304). 2.
Definition
and elementary properties of Wick and Anti-Wick symbols
(307).
3. Covariant and contravariant symbols (316). 4. Convexity
inequali-
ties and Feynman-type inequalities (321).
5.3. The General Concept of Feynman Path Integral in Phase
Space.
Symbols of the Evolution Operator . . . . . . . . . . . . . .
324
1. The method of Feynman Path integrals (324). 2. Weyl symbol
of
the evolution operator (328). 3. The Wick symbol of the
evolution
operator (345). 4. pq- and qp-symbols of the evolution operator
and
the path integral for matrix elements (357).
5.4. Path Integrals for the Symbol of the Scattering Operator and
for the
Partition Function 361
x CONTENTS
1. Path integral for the symbol of the scattering operator
(361).
2. The path integral for the partition function (370).
5.5. The Connection between Quantum and Classical Mechanics.
Semi
classical Asymptotics . . . . . . . . . . . . . . . . . . . .
374
1. The concept of a semiclassical asymptotic (374). 2. The
operator
initial-value problem (374).3. Asymptotics of the Green's function
(377).
4. Asymptotic behaviour of eigenvalues (381). 5. Bohr's formula
(383).
SUPPLEMENT 1. Spectral Theory of Operators in Hilbert Space
386
S1.1. Operators in Hilbert Space. The Spectral Theorem . . . . . .
. 386
1. Preliminaries (386). 2. Theorem on the spectral decomposition of
a
self-adjoint operator in a separable Hilbert space (392). 3.
Examples
and exercises (406). 4. Commuting self-adjoint operators in
Hilbert
space, operators with simple spectrum (407). 5. Functions of
self
adjoint operators (411). 6. One-parameter groups of unitary
operators
(414).7. Operators with simple spectrum (415). 8. The
classification
of spectra (416). 9. Problems and exercises (418).
S1.2. Generalized Eigenfunctions . . . . . . . . . . . . . . . . .
419
3. Rigged Hilbert spaces (423). 4. Generalized eigenfunctions
(426).
5. Statement and proof of main theorem (429). 6. Appendix to
the
main theorem (430). 7. Generalized eigenfunctions of differential
op
erators (431).
Spectrum . . . . . . . . . . . . . . . . . . . . . . . . .
434
S1.4. Trace Class Operators and the Trace . . . . . . . . . . . . .
448
1. Definition and main properties (448). 2. Polar decomposition of
an
operator (451). 3. Trace norm (453).4. Expressing the trace in
terms
of the kernel of the operator (457).
S1.5. Tensor Products of Hilbert Spaces . . . . . . . . . 462
SUPPLEMENT 2. Sobolev Spaces and Elliptic Equations 466
S2.1. Sobolev Spaces and Embedding Theorems. . . . . . 466
S2.2. Regularity of Solutions of Elliptic Equations and a priori
Estimates 475
S2.3. Singularities of Green's Functions . . . . . . . 480
SUPPLEMENT 3. Quantization and Supermanifolds 483
S3.1. Supermanifolds: Recapitulations . . . . . . . . 486
bras (499). 3. Lie supergroups and homogeneous superspaces in
terms
of the point functor (507). 4. Two types of mechanics on
supermani
folds and Shander's time (509).
S3.2. Quantization: main procedures. . . . . . . . . . . . . . . .
511
S3.3. Supersymmetry of the Ordinary Schrodinger Equation and of
the
Electron in the Non-Homogeneous Magnetic Field 520
A Short Guide to the Bibliography
Bibliography
Index ....
523
533
551
Foreword
The Schrodinger equation is the basic equation of quantum theory.
The study
of this equation plays an exceptionally important role in modern
physics. From
a mathematician's point of view the Schrodinger equation is as
inexhaustible
as mathematics itself.
In this book an attempt has been made to set forth those topics of
math ematical physics, associated with the study of the
Schrodinger equation, which
appear to be the most important.
Intended mainly for students of mathematics, the book starts with
an in
troductory chapter dealing with the basic concepts of quantum
mechanics. This
would help the reader well versed in mathematics to understand the
physical
meaning of the mathematical constructions and theorems expounded in
the
subsequent chapters. One should not think that this concise chapter
can serve
as a substitute for a systematic study of physical textbooks on
quantum me
chanics. It is hoped, however, that the perusal of the book would
be sufficient
for a mathematician to take in these textbooks.
The point is that current textbooks on quantum mechanics are
mainly
intended for physicists * and present considerable difficulties to
students of mathematics. This is associated with the fact that a
systematic presentation
of the general concepts of quantum mechanics requires rather
extensive prelim
inary knowledge of general functional analysis (spectral theory of
operators,
the concept of a generalized eigenfunction, etc.), partial
differential equations,
and some other advanced elements of mathematics. An attempt to skip
this
information by substituting it with a reference to similar
information taken, for
* An exception is the book by Faddeev and Yakubovskii [1].
xiii
xiv FOREWORD
example, from finite-dimensional linear algebra, would look like
regular cheat
ing to the reading mathematician unless he is inclined to read
between the
lines. The authors have tried to avoid creating this kind of
impression in the
present book.
The reader is not supposed to possess any mathematical knowledge
beyond
the scope of conventional calculus, supplemented by elementary
information on
the theory of distributions. However, a relatively standard part of
the informa
tion presented here is given in the Supplements 1 and 2 which
should be studied
systematically by the reader unfamiliar with it, whereas a more
advanced reader
can use them when required. This has enabled the authors to begin
the book
with the postulates of quantum mechanics, that is, its mathematical
scheme.
This book is based on the lectures given more than once by the
authors at
the Department of Mechanics and Mathematics of the Moscow State
Univer
sity and in part published in 1972 (see Berezin and Shubin [2]).
Though the
lectures were meant for students ofmathematics, the experience with
the above
mentioned preliminary publication showed that the type of material
presented
here can also be useful to the physicists who want to familiarize
themselves
better with the mathematical formalism of quantum mechanics.
It should be emphasized that the authors have in no way tried to
attain
any completeness, being aware that no book of reasonable volume can
exhaust
the subject to any great extent. * The book, however, covers many
important ideas and a number of profound theorems. We hope,
therefore, that the reader
who has mastered the content of the book will be ready, or almost
ready, to
work actively in the field of mathematics and mathematical physics
discussed
here.
The mathematical theorems treated in the book are not new,
although
most of the proofs differ from the more familiar ones. The authors
have tried
to present in more detail the results that cannot be found in
monographs. At
the same time, the results repeatedly treated in easily available
books are often
described briefly, almost as a synopsis (in these cases the
necessary references
are given). Bibliography has been reduced to a minimum, the
monographs and
* As is well known, "... of making books there is no end; and much
study is a weariness of the flesh" - this conclusion of
Ecclesiastes (12:12) was also
reached by Read and Simon ([1], (vol. 4» after they had written
four volumes
of their course on modern mathematical physics.
FOREWORD xv
papers referred to being only those most closely connected with the
text.
In some places, the authors took the liberty of shifting from the
exact
mathematical language to the heuristic one in the belief that the
knowledge of
the key ideas which lead to the correct answer is often more
important than
tiresome details. In some cases though (particularly in the theory
of Feynman
path integrals) the authors were forced to make such a shift, since
an accurate
mathematical substantiation is unknown or insufficient for the
given problem.
Let us now describe the content of the book in somewhat greater
detail.
Chapter 1 describes the mathematical structure of quantum
mechanics. In
principle, a similar scheme functions in physics, although it is
seldom formu
lated so explicitly. At the same time, this scheme should not be
regarded as a
dogma but as a guide to action, although there is already an
infinite wealth of
content within its framework.
Chapter 1 is based on the mathematical material given in §1 and §2
of
Supplement 1.
Chapter 2 is an introduction to the spectral theory of the
one-dimensional
Schrodinger operator. The main problems considered include
conditions of
essential self-adjointness, the nature and structure of the
spectral behaviour
of eigenfunctions, expansion in terms of eigenfunctions, the
inverse scattering
problem, and Bloch eigenfunctions of operators with periodic
potentials.
Chapter 2 makes use of the material presented in Supplement
1.
Chapter 3 opens with a discussion of the spectral theory of the
multi
dimensional Schrodinger operator. At the beginning, the problems
treated
are essentially the same as those set forth in Chapter 2, but for
the multi
dimensional case. The latter is, however, far more complex and the
technique
developed in the theory of partial differential equations (Sobolev
spaces and
regularity of solutions for elliptic equations) has to be applied
here. The nec
essary preliminaries are described briefly in Supplement 2.
Moreover, new
subtle points arise which are trivial in the one-dimensional case
(for example,
the question concerning the existence of eigenvalues immersed in a
continuous
spectrum in the case of the operator with decaying
potential).
Chapter 4 presents the scattering theory for the multi-dimensional
non
relativistic Schrodinger equation. Specifically, we build the
complete system of
generalized eigenfunctions for the Schroodinger operator with
decaying poten
tial. We discuss the non-stationary approach by Enss as well as a
variant of
xvi FOREWORD
the stationary approach.
Chapter 5 is devoted to quantization, or the theory of symbols as
well as
to Feynman path integrals. All this is presented for the simplest
case of a linear
phase space. Here we obtain different asymptotics (including the
semi-classical
ones). But the content of this chapter is now predominantly of a
heuristic
nature, that is, the formulas and statements presented are, for the
most part,
not proved mathematically but only derived on the physical level of
rigour.
The reasons, making such deviation from mathematical rigour
inevitable, have
already been mentioned above.
This edition contains more material than the Russian edition of
1983. In
particular, a new Supplement 3 is added. This chapter stands apart.
A new
concept, supermanifold, is introduced and applied to a more natural
treatment
of quantization there. Besides, in Supplement 3, several different
approaches
to quantization close in spirit to Berezin's (the latter included)
are discussed.
Supplement 1 contains general questions of the spectral theory of
operators
in Hilbert space.
In Supplement 2, we present the necessary information on Sobolev
spaces
and elliptic equations.
The demands imposed on the reader are growing with the numbers
of
chapters (this is not true for Supplements 1 and 2, intended for
beginners).
Not all the chapters, however, rest on all the preceding material.
The diagram
of the interrelation of the chapters and the supplements is given
below. Us
ing the diagram, the reader will be able to choose the route he
prefers if he
is not interested in all the material of the book. Note, for
instance, that, in
principle, only Chapters 2 to 4 may be read as an introduction
(together with
Supplements) to the spectral theory of the Schrodinger operator. It
is helpful,
however, to start this route from Chapter 1, otherwise the spectral
theory, rich
in its physical content, could turn in the reader's eyes into a
purely intellec
tual sport. Another route - Chapter 1, 5 and Supplement 3 - may
attract
physicists, as well as mathematicians who are looking at
mathematical physics
as an area for applying their abilities.
Lest the reader's attention should be diverted, we have given
almost no
references in the main text. These are presented at the end of the
book.
Unfortunately, this book had to be prepared for publication without
its
principal author - Felix Aleksandrovich Berezin, whose tragic death
prevented
FOREWORD XVII
him from actualizing his intentions in this book (as well as many
other projects).
The outline of the book drawn up by F.A. Berezin contains a number
ofsections
that could have been written only by him and so remained unwritten.
His
research has played an extremely important role in the development
of modern
mathematical physics. F .A. Berezin was the founder of the
supermanifold
theory, the tool that provides us nowadays with the first
divergence-free model
of the unified field theory. F.A. Berezin was a remarkable person.
I learnt a
great deal from F.A. Berezin, feel highly indebted to him, and am
well aware
of how much better this book could have been had F.A. Berezin
himself taken
part in its completion.
Numerous colleagues and friends gave me their valuable assistance
in the
work on this book. Thus, G.L. Litvinov wrote the first chapter, as
well as
subsections 4-6 of §1 and §5 of Supplement 1, and D.A. Leites wrote
Supple
ment 3. L.A. Bagirov told me the proof of the Kato theorem
presented in §4 of
Chapter 3, A.S. Schwarz helped me in composing the final outline of
the book
and inspired me to bring it to completion. A number of useful
discussions
were held by the authors with V.P. Maslov and A.I. Naumov. L.M.
Ioffe and
A.M. Stepin wrote down the first course of F.A. Berezin's lectures.
I wish to
express my deep gratitude to all of them. I am also grateful to my
mother for
invaluable and selfless help in preparing the manuscript, as well
as to all the
other members of my family for their patience.
M.A. Shubin
Chapters and Supplements
----1~~ Chapter 3
Chapter 1 I I
+ Supplement 3
Notational Conventions
Z, Z+, R, R+, and C denote, as usual, the sets of integers,
non-negative
integers, real, non-negative real, and complex numbers
respectively; Cij = 0 for i :f j and 1 for i =j. Reference to
Theorem S1.5.2 or Lemma 2.1.5 is to Theorem 2 from §5 of
Supplement lor Lemma 5 from §1, Chapter 2. Inside a chapter,
notations are
shortened to Theorem 5.2 or Lemma 1.5 respectively. Reference to
§S1.5.2 is
to subsection 2 of §5 of Supplement 1.
CHAPTER 1
Introduction
The starting point in the development of quantum mechanics was
marked by
the work of M. Planck on radiation theory, published in 1900. The
fact is
that the application of the principles of classical physics to the
analysis of
the spectral distribution of thermal radiation energy leads to the
"ultraviolet
catastrophe": the density of energy of equilibrium radiation
becomes infinitely
high. This means that at any temperature, thermal equilibrium
between matter
and radiation is impossible because the matter should radiate
energy until it
is cooled to absolute zero. In order to obtain a radiation energy
distribution
law conforming to experiment Planck had to assume electromagnetic
radiation
to be emitted and absorbed in separate portions - quanta, the
energy E of a
quantum being proportional to the circular frequency of radiation
w:
E=hw,
where h = 1.05· 1O-27erg·sec. The constant h is known as Planck's
constant*.
* In the works of the founders of quantum theory, Planck's formula
was written as E = hv, where v = w /27r is the conventional
oscillation frequency of
an electromagnetic wave. Correspondingly, in old literature
Planck's constant
was
1
2 CHAPTER 1
By assuming that light is not only discretely emitted and absorbed
but
also propagated as discrete quanta (photons), A. Einstein was able
to explain
in 1905 the laws of the photoelectric effect. Einstein attributed
to each quantum
of light not only the energy, in accordance with Planck's formula,
but also the
momentum vector, whose length p is related to the light wavelength
A by
27rh P=T'
Einstein's hypothesis was experimentally confirmed in 1923 by A.
Comp
ton who showed that photon-electron collisions comply with the
energy and
momentum conservation laws in accordance with Planck's and
Einstein's for
mulas. In 1924 L. de Broglie formulated his hypothesis that these
relations
reflect the universal wave-particle duality. In particular, de
Broglie associated
with the motion of any particle a wave of length A= 27rh/p, where p
is the
length of the particle's momentum vector. Wave properties of
microparticles
were later revealed by C. Davisson and A. Jarmer in their
experiments on
electron diffraction on crystal lattices (1927) and by other
experiments.
The planetary model of atom, proposed and experimentally
substantiated
by E. Rutherford (1911), also contradicts the fundamentals of
classical physics.
According to classical electrodynamics, electrons moving around the
nucleus
along closed orbits must, as any other accelerated charges, radiate
electromag
netic waves. As a result, electrons, losing their energy, must fall
on the nucleus
within a time of the order of 10-9 sec (which, of course, does not
take place in
actual fact). Moreover, according to classical mechanics, an
electron can move
along any orbit and, therefore, emit light of any wavelength. It
is, however, well
known that radiation spectra of many substances are discrete. To
explain the
structure of atoms, N. Bohr proposed in 1913 a theory combining the
principles
of classical physics and the additional postulates contradicting
them. In partic
ular, Bohr postulated the existence of stationary orbits, moving
along which an
electron does not radiate, its energy values being only discrete.
Transitions of
an electron from one stationary orbit to another are accompanied by
emission
or absorption of photon, its energy being determined by the
difference in the
energies of the corresponding orbits. This theory, supplemented and
perfected
by A. Sommerfeld and other authors, is often referred to as "the
old quan
tum theory". The old quantum theory, which was regarded by Bohr as
only
a step in the search for a more correct and consistent theory, made
it possible
GENERAL CONCEPTS OF QUANTUM MECHANICS 3
to explain qualitatively the structure of atomic spectra and give a
quantitative
description of the properties of the hydrogen atom and one-electron
ions. This
theory, however, was unable to explain the properties of more
complex atoms
and molecules.
A consistent theory of microscopic phenomena is the quantum
mechanics
developed by W. Heisenberg, E. SchrOdinger, M. Born, P. Jordan, N.
Bohr, W.
Pauli, P.A.M. Dirac and other scientists. In 1925 Heisenberg
outlined a new
approach to the theory of atomic phenomena. Following it, Born,
Heisenberg
and Jordan developed matrix mechanics where physical quantities are
repre
sented not by numbers or numerical functions but by infinite
matrices. In 1926
Schrodinger, developing de Broglie's ideas, constructed the wave
mechanics,
within the framework of which, the determination of the values of
physical
quantities was reduced to the calculation of the eigenvalues of
linear differen
tial operators. After Schrodinger and other authors had established
that the
matrix and the wave mechanics constituted two possible ways of
presenting the
same theory, this theory became known as quantum mechanics. The
develop
ment of the formalism of non-relativistic quantum mechanics had
been mainly
completed by the end of the twenties, but the questions of its
foundations and
interpretation continued to be actively discussed in the years that
followed. A
rigorous mathematical justification of the formalism of quantum
mechanics was
elaborated by J. von Neumann [1] in the late twenties and early
thirties.
In this chapter we give a logical and mathematical scheme of
non-relativ
istic quantum mechanics in the spirit of von Neumann's ideas. Such
concepts as
distribution and generalized eigenvector (introduced into
mathematical prac
tice in the fifties) provide us with a well-defined mathematical
interpretation of
"singular" objects like Dirac's delta-function, systematically used
in the phys
ical literature. The deductive method of exposition which we follow
requires
some patience from the reader since the rather abstract notions and
construc
tions are illustrated by examples only after §3. Some questions
examined in
detail in most of the standard textbooks on quantum mechanics are
formulated
as exercises or omitted; in particular, the description of the
hydrogen atom is
omitted. There are no descriptions of experiments. The history of
quantum
mechanics is touched upon only occasionally and the names of the
authors of
certain concepts and results are usually mentioned only when
required by tra
dition. There exists a lot of literature, including popular books,
that can help
4 CHAPTER 1
the reader to fill in all these gaps. Some literature is cited in
the commentary
to Chapter 1 at the end of the book.
1.1. Formulation of Basic Postulates
Quantum mechanics is used to study micro-objects, such as atoms and
mole
cules, and processes whose physical characteristics are of the
order of Planck's
constant. For macro-objects, that is, objects of usual size, whose
behaviour is
adequately described by classical mechanics, Planck's constant can
be regarded
as negligibly small.
Human sense organs are unable, as a rule, to perceive
microphenomena
directly. Therefore, to investigate the behaviour of micro-objects
we need an
intermediary apparatus (not necessarily man-made) whose behaviour
is de
scribed by classical mechanics.
In order to measure a physical quantity of a micro-object, we must
make
it interact with the apparatus. As a result of this interaction,
the macroscopic
state of the apparatus changes, that is, the act of measurement
takes place.
Interacting with the apparatus a micro-object usually causes an
avalanche-like
process (for example, vapour condensation in a cloud chamber),
which leads to
a change in the state of the apparatus.
It so happens that, in general, one cannot predict the exact result
of a
measurement (even when one has all the possible information on the
condi
tions in which the measurement is performed). The result of a
measurement
is a random variable, and quantum mechanics deals with the
probability dis
tributions of such variables. Physical quantities whose values can
be (if only
in principle) determined experimentally are called observables. The
results of
measurement are supposed to be real numbers.
The system of basic postulates of quantum mechanics, described
below,
was proposed by J. von Neumann [1]. For the mathematical concepts
and
results used, see Supplement 1.
Postulate 1. The states of a quantum mechanical system are
described by
non-zero vectors of a complex separable Hilbert space £, two
vectors describing
the same state if and only if they differ only by a non-zero
complex factor. Each
observable corresponds to a certain (unique) linear self-adjoint
operator in £.
GENERAL CONCEPTS OF QUANTUM MECHANICS 5
The space £, is called the state space, and the elements of this
space - the
state vectors. We always assume (unless otherwise specified) that
the vector
"p E £', describing the state of a physical system, has unit
length. In some cases,
meaning that the state of a system is described by a vector "p E
£', we say that
the system is in the state"po The operator corresponding to an
observable
a will be denoted by a. Observables aI, a2, ... ,an are called
simultaneously
measurable (or jointly measurable) if their values can be measured
with an
arbitrary accuracy in one experiment so that the random variables
aI, . 0 • , an
have a joint distribution function P", (AI,. 00' An) for an
arbitrary given state
"p.* Let al, ... ,an be simultaneously measurable observables. In
other words,
Pt/J (AI,.' 0' An) with fixed values of the arguments is the
probability of the
values of the observables al, ... , an measured at "p to be not
greater than
AI,' .. , An respectively.
Postulate 2. Observables are simultaneously measurable if and only
if the
corresponding self-adjoint operators commute. If observables aI,
... ,an are si
multaneously measurable then, for a given "p, their joint
distribution function
is of the form
(1.1)
where Ei~), ... ,Et) are the projection operators ** of the
spectral families
corresponding to the operators aI, 0 •• ,an 0
It is clear that the value of (1.1) does not change if the vector
"p is re
placed by another vector representing the same state of the system
(it is to
be remembered that states are described by normalized vectors).
Since the
Ei~),. 0" Et) commute, the value of (1.1) does not depend on the
sequence in which the observables aI, . 0 • ,an are considered. If
anyone of the observables,
say a, is measured, the corresponding distribution function at the
state "p is
(1.2)
where E>. is the projection of the spectral family of a; (1.2)
is a special case of (1.1).
* The question of simultaneous measurability of observables in
connection with the Heisenberg uncertainty relations is discussed
in detail in §5. ** For the sake of brevity we shall write
"projection" instead of "projection operator" below.
6 CHAPTER 1
The most important physical quantity in any quantum mechanical
system
is its energy. The corresponding operator will be denoted by H. The
follow
ing postulate asserts that the energy operator * H determines the
law of the
system's evolution.
Postulate 3. Let the state of a system at t = 0 be represented by a
vector
'if;o. Then at any time t the state of the system is represented by
the vector
'if;(t) = Ut'if;o, where Ut is a unitary operator called the
evolution operator. The
vector-function 'if;(t) is differentiable if'if;(t) is contained in
the domain DH of
H (if only at t = 0) and in this case:
ih d~~t) = H'if;(t), (1.3)
where h is Planck's constant and i = A, the imaginary unit.
Relation (1.3) is the basic equation of quantum mechanics and is
called
the Schrodinger equation.
Henceforth, unless otherwise specified, we assume that the energy
op
erator H (as well as other operators corresponding to observables)
is time
independent. In this case the main statement of Postulate 3
becomes:
Evolution operators Ut constitute a strongly continuous
one-parameter group,
generated by the operator -kH, that is,
(1.4)
In fact, from the properties of such groups (see §Sl.1.6) and from
(1.4), it
follows that 'if;o E DH if and only if the vector-function 'if;(t)
is differentiable at
t =0 and d~~t) It=o = -kH'if;o. In this case, for all t, there
exists a derivative
of the vector-function 'if;(t) and 'if;(t) E DHj indeed,
d'if;(t) =lim Ut+.'if;o - Ut'if;o =lim U.'if;(t) - 'if;(t) = dt
.-+0 s .-+0 s
d i = dsl.=o (U.'if;(t)) = -hH'if;(t).
We have thus arrived at the Schrodinger equation. As DH is dense in
£, the group Ut is uniquely determined by the Schrodinger
equation.
* The energy operator is often referred to as the Hamiltonian. In
non relativistic quantum mechanics it is also called the
Schrodinger operator.
GENERAL CONCEPTS OF QUANTUM MECHANICS 7
It should be borne in mind that the relations (1.3) and (1.4)
remain valid
only as long as the system is not exposed to an external
perturbation. An ex
ample of such a perturbation that cannot be disregarded is an act
of measuring
an observable.
Postulate 4. Every non-zero vector of the space {, corresponds to a
state of
the system and every self-adjoint operator corresponds to an
observable.
Postulate 4 implies the so-called superposition principle: if a
system can
be in states described by vectors tP1 and tP2, then it can also be
in any state described by their superposition, that is, an
arbitrary linear combination of
them.
The question of how, for a concrete physical system, to describe
the state
space {, and establish a correspondence between the observables and
self-adjoint
operators in £ goes beyond a purely mathematical theory and belongs
to the
domain of physical practice and intuition. * In every particular
case this ques tion should be handled by an expert
physicist.
One should not think that von Neumann's axioms constitute the only
pos
sible or final way to describe the basic concepts of quantum
mechanics. We have
to introduce additional postulates when analyzing specific systems,
for exam
ple systems of identical particles. Though for the majority of
physical systems
considered in this book Postulate 4 is valid, in the early fifties
the superpo
sition principle and, consequently, Postulate 4, were found not to
be always
true. There exist the so-called superselection rules, which split a
state space
into a direct sum of orthogonal subspaces. The rules then state
that the sum of
non-zero vectors from different subspaces cannot correspond to any
physically
realizable state. It is impossible, for example, to realize
superposition of states
corresponding to different values of electric charge or baryon
number. The
operators of observables must commute with the projections onto
subspaces,
singled out by superselection rules, that is, leave these subspaces
invariant.
Hence, instead of Postulate 4, it is advisable to adopt the more
general
* Although all the Hilbert spaces of a fixed dimension are
isomorphic, the state space of a concrete physical system has
important additional structures
related to properties of the energy operator, the existence of a
natural rigging,
etc.
8 CHAPTER 1
Postulate 4'. In the space £, there exists a family of mutually
commuting self
adjoint operators {Pa,} such that the observables correspond to
those and only
those self-adjoint operators in £, which commute with all the
operators of this
family.
It is clear that Postulate 4 is valid in the special case when the
family {Pal
is empty or contains only scalar operators. In the general case, it
is said that
the operators Pa determine superselection rules. Superselection
rules require
that every observable can be measured simultaneously with all the
observables
corresponding to the operators Pa .
A vector t/J E £, is considered to represent a "physically
realizable" state if the orthogonal projection operator onto the
straight line generated by t/J commutes with the Pa's, that is,
when this projection corresponds to a certain
observable. It is also natural to believe that, if a state t/J is
physically realizable, its energy is finite. We show below that, if
t/J is contained in the domain of the energy operator, the energy
of the state t/J has a finite mean value and a finite variance.
Since commutability with a self-adjoint operator is
equivalent
to commutability with the corresponding spectral projectors,
superselection
rules can be determined by sets of projectors. If linear
combinations of vectors
representing physically realizable states are dense in £', then
Postulate 4' im
plies that any superselection rule can be prescribed by a set of
projectors onto
mutually orthogonal subspaces; in this case the space £, splits
into the sum of
them. The proof of this statement is offered to the reader as an
exercise. * For different variants of axiomatic description of the
quantum theory prin
ciples see Comments to Chapter 1.
1.2. Some Corollaries of the Basic Postulates
Let a be an arbitrary observable, a the corresponding self-adjoint
operator with the domain Do.; the domain of any operator A is
further denoted by DA. The
* A theoretical case is imaginable, when the {Pa } determine a
decomposition
of the state space into an integral, instead of a sum, so that no
state vector
is "physically realizable". Then only physically valid states are
mixed ones,
which are determined by density operators commuting with all the
Pa's (cf.
§2, Exercise 6).
GENERAL CONCEPTS OF QUANTUM MECHANICS 9
mean value (the expectation) of an observable a in a state tf; will
be denoted
by a",.
Proposition 2.1. If tf; E Da, then the mean value of a in the state
tf; is given
by the formula
(2.1)
Proof It follows from the theorem on spectral decomposition of a
self-adjoint
operator (see Theorem S1.1.1') that (atf;,tf;) =
f~oo>'d(E>..tf;,tf;), where E>.. is
the spectral family corresponding to the operator a. As E~ =
E>.. (property of
a projection) and the operators E>.. are self-adjoint,
therefore, from the relation (1.2) it follows that
This is just what we had to prove as f~oo >'dP",(>.) is
nothing but the
expectation of a random variable with the distribution function
P",(>.). •
Let f(>') be a real-valued function that is measurable and
almost every
where finite with respect to the measure d( E>.. tf;, tf;), for
example continuous.
Then the self-adjoint operator 1(0.) is defined, (see §S1.1.5). Let
us denote by
f(a) the observable taking the value f(>.) when the observable a
assumes the
value >..
Proposition 2.2. If tf; E Dj(a), then the mean value of f(a) in the
state tf; is
determined by the formula:
[f(a)]", = (J(a)tf;, tf;). (2.2)
Proof As is known from probability theory (and is easily deduced
from the
definitions), the left-hand side of equality (2.2) coincides with
f~oo f(>')dP",(>.) ,
where P",(>.) is the distribution function ofthe observable a in
the state tf;. It
remains to note that
10 CHAPTER 1
by Proposition 51.1.12. •
It follows from Proposition 2.2 that the self-adjoint operator
corresponding
by Postulate 1 to the observable f(a) coincides with f(a) (see
Exercise 2 below).
The dispersion of the values of a random variable relative to the
mean value
is characterized by variance. We denote by 6",a the variance of an
observable
a in a state t/J, that is, the mean value of (a - a",)2.
Proposition 2.3. The variance of an observable a in a state t/J
exists if and
only if t/J E Da. In this case it is defined by
(2.3)
Proof. In a state t/J the variance of an observable a exists if and
only if the
integral
converges. It is easy to see that convergence of this integral is
equivalent to
the convergence of J~oo>..2d(E>.t/J,t/J),which, by Theorem
51.1.1', means that
t/J E Da. From Proposition 2.2 and the self-adjointness of the
operator a-a",I,
where I is the identity operator, it follows that
6",a =((a - a", . I) 2 t/J, t/J) =((a - a",I)t/J, (a - a", . I)t/J)
=
= 110,,,, - a",t/Jlr~·
Proposition 2.4. In a state t/J an observable a takes a value>"
with certainty
(that is, with probability 1) if and only if t/J is an eigenvector
of the operator a with the eigenvalue >...
Proof. If in a state t/J an observable a takes the value >..
with certainty, then 6",a = 0, a", = >... Therefore it follows
from formula (2.3) that 110,,,, - >..t/JII =0, that is, at/J
=
>..t/J. Then from formula (2.1) it follows that a", = (at/J,
t/J) = >..('ljJ, 'ljJ) = >.., so 6",a = lIit'ljJ -
>..t/J1I 2 = o. • Of particular importance is the case, when the
state of a physical system
is described by an eigenvector of the energy operator. Let the
state of the
system at t = 0 be described by an eigenvector 'ljJo of the energy
operator
GENERAL CONCEPTS OF QUANTUM MECHANICS 11
H with the eigenvalue AO' The vector-function 'I/;(t) =
e-i>.ot/h'l/;o satisfies the
Schrodinger equation (we recall that the operator H is considered
to be time
independent). The vector 'I/;(t) differs from '1/;0 only by a
numerical factor and,
consequently, describes the same state of the system. Now assume
that Ut'l/;o = c(t) '1/;0 , where Ut is the evolution operator and
c(t) a numerical factor. Since
c(t) = (Ut'l/;o, '1/;0), this function is continuous; the relation
c(t + s) = c(t)c(s)
follows from the group property UtU. = Ut+• . It is well known that
continuous
functions satisfying this functional equation are exponentials.
Hence c(t) is
differentiable. Thus,
H'I/;o = ih d d
Ut'l/;0l = AO'l/;o, where AO = ih dcd(t) I . t t=O t t=O
We have proved the following
Proposition 2.5. A state of a system is time-independent if and
only if it is
represented by an eigenvector of the energy operator.
A time-independent state is called stationary state. Energy in a
stationary
state with certainty takes one value - the eigenvalue of the energy
operator.
The equation H'I/; = A'I/; describing the stationary states
(eigenvectors of
the energy operator H) is often referred to as the time-independent
Schrodinger
equation.
Let a be an arbitrary observable, a the self-adjoint operator
corresponding to it, E the projections of the spectral family of
the operator a. If ~ is a set of
real numbers measurable with respect to the measure d(E>. '1/;,
'1/;) for any 'I/; E .c, then we denote by h(A) the characteristic
function of the set ~, taking the value 1 when AE ~ and 0 when Afi.
~. Set
E(~) = h(a).
The operator E(~) is self-adjoint and (E(~))2 = E(~), because f1 =
h.
Hence, E(~) is an orthogonal projection; if the sets ~1 and ~2 are
disjoint
then E(~1)E(~2)=h,ftl2(a) =0, that is, the corresponding subspaces
are orthogonal. If ~ is a half-open interval, (Al, A2] consisting
of A such that
A1 < A S A2' then E(~) = E>'2 - E>'l; for A1 = -00, >'2
= A the operator
E(~) becomes the projection E>. of the spectral family. Finally,
if ~ contains
only one point A, then E(~) = E>. - E>.-o, where E>._o is
the limit of the operators E>._. as f. -. 0 remaining
positive.
12 CHAPTER 1
Proposition 2.6 The probability for the value of an observable a
measured in
a state 1/; to belong to ~ is IIE(~ )1/;112.
Proof. This probability equals
= (E(~)1/;, E(~)1/;) = IIE(~)1/;1I2. •
From Proposition 2.6 it follows that the observable a with
certainty takes
the values belonging only to ~ exactly when 1/; belongs to the
range £/j" = E(~)£ of the projection E(~). If ~ consists of a
single eigenvalue>. of an
operator ii, then by Proposition 2.4, the subspace £/j" = (E>. -
E>.-o)£ consists
of all the eigenvectors of ii with the eigenvalue >.. If a
number >. is not the
eigenvalue of a, but E(~) i 0 for any interval ~, containing the
point >., then, although there is no state in which the
observable a with certainty takes the
value >., one can find states where this observable takes values
differing from >.
by an arbitrarily small amount.
Let 1/; = J,"f(m)S(m)d(J(m) be the expansion of a vector 1/; in
generalized
eigenvectors S(m) of a (see §S1.2). If ,"f(m) i 0 only when the
eigenvalue of the generalized vector S(m) belongs to ~, then 1/; E
£/j". Thus, if 1/; is a
"continuous linear combination" of the generalized vectors of the
operator ii,
whose eigenvalues slightly differ from >., then the values taken
by the observable
a in the state 1/; also differ slightly from >.. Therefore, it
is often said, by abuse of
language, that the generalized eigenvector of the operator awith
the eigenvalue >. represents a state of the system in which the
observable a reliably takes the
value >..
Let us consider several examples.
1. Let abe the orthogonal projection onto a certain subspace of £.
Then the observable a can take only one of the two eigenvalues of
a, that is 0 or 1. It is easy to see that in the state 1/; the
observable a takes the value 1 with
probability 110.1/;112 and the value 0 with probability II1/; -
a1/; II 2.
2. If a is the orthogonal projection onto a one-dimensional
subspace gen erated by a unit vector 1/;0 E £, then the observable
a takes the value 1 with
certainty if and only if the quantum mechanical system in question
is in the
GENERAL CONCEPTS OF QUANTUM MECHANICS 13
state 1/;0 (see Proposition 2.4). If, however, the system is in an
arbitrary state
1/;, the observable a takes the value 1 with probability
I(1/;,1/;0)12 and the value owith probability 1-1(1/;,1/;0)12. 3.
Let an observable a be such that the operator a has a simple
pure
point spectrum, that is, there exists an orthonormal basis, 1/;1,
1/;2,· .. ,1/;i, ...
in the state space £, such that a1/;i = Ai1/;i, where Ai are
numbers such that
Ai 'f; Aj for i 'f; j. Let us denote by Qj the orthogonal
projection onto the
one-dimensional space generated by 1/;j. Then the projections of
the spectral
family corresponding to the operator a have the form E>. =
L>.<>. Qj. If J-
1/; = Li ci1/;i is the expansion of a unit vector 1/; E £, in the
basic vectors,
then in the state 1/; the observable a takes the value Ai with
probability Pi = IIQi1/;1I2 = 1(1/;,1/;iW = lcil 2 • In this case,
the Parseval equality implies that Li Pi =1. The mean value and the
variance of the observable a in the state 1/;
are:
a1/l = LAiPi = L Ailcil2; 81/1a = L(a,p - Ai)2IciI2. iii
If 81/1 a = 0 then a1/l = >'k for a certain k and, all the
eigenvalues of the operator
a being different, Ci = 0 for i f:: k. Consequently, if the
observable a with
certainty takes the value AI:, the physical system undoubtedly is
in the state
1/;1: .
Exercises
1. Let the space L2(R) offunctions 1/;(x) on the real axis
square-integrable
with respect to the Lebesgue measure be the state space and let the
operator
a : 1/;(x) -+ x1/;(x) correspond to an observable a. Describe the
distribution
function of this observable in an arbitrary state.
2. If a is an arbitrary observable and A is the self-adjoint
operator corre
sponding to the observable /(a) according to Postulate 1, then A =
/(a).
Hint. If / is bounded, then according to Proposition 2.2, we have
(A1/;, 1/;) =
(/(o')1/;, t/J) for all1/; E£'. Derive from it that (A1/;1l1/;2) =
(/(0,)1/;1,1/;2) for all
1/;1, 1/;2 E£'. For the general case, establish that the operators
A and /(a) have
the same spectral family.
3. From Exercise 2 and Proposition 2.1 derive Proposition 2.6 and
equa
tion (1.2).
4. From Exercise 2, Proposition 2.1 and Postulate 4 derive equation
(1.1).
Generalize this result to superselection rules.
14 CHAPTER 1
Hint. There exists a self-adjoint operator A and functions
ft()..),··· ,fn()..) such that 0.1 = ft(A), .. . , an = fn(A). If
superselection rules are determined
by a set of projections Pa , then one can choose A so that the
projections Pa
become functions of A and hence commute with A (see §S1.1.5).
5. Prove that the set of values of an observable a coincide with
the spec
trum of the operator a. Hint. Prove that ).. does not belong to the
spectrum of a if and only if
E(Ll) = 0 for some open interval Ll containing the point )..; use
Proposition 2.6.
6. Let 1/Jl' 1/J2, ... ,1/Jn, . .. be mutually orthogonal
normalized vectors in the state space, and suppose that the system
is in the state 1/Jn with probability Pn, where I:Pn = 1. In this
case the system is said to be in a mixed state. It is
convenient to characterize the mixed state by the density operator
introduced
by von Neumann. The density operator has the form
n
Therefore 1/Jn is the eigenvector of the operator P with the
eigenvalue Pn. Prove that P is a non-negative trace class operator
* and that the expectation (mean value) a of the observable a in
the corresponding mixed state is given by the
formula
a = tr(ap).
This formula turns into (2.1) if P coincides with the orthogonal
projection
onto the one-dimensional space generated by a normalized vector
1/J; in this case P is said to characterize a pure state. Prove
that the density operator
P characterizes a pure state if and only if P cannot be represented
as a sum
of non-trivial non-negative trace class operators. Describe the
evolution of a
mixed state in terms of the density operator. Prove that any
non-negative trace
class operator in the state space I:- is the density operator for a
certain mixed
state.
1.3. Time Differentiation of Observables
Consider an arbitrary observable a. Even if the operator a is
time-independent
* See §S1.4 concerning trace class operators and tra.ces.
GENERAL CONCEPTS OF QUANTUM MECHANICS 15
(only this case will be considered), the distribution function of
the correspond
ing random variable is time-dependent, since the state vector is
time-dependent.
Suppose that at t = 0 the state of a physical system is represented
by a fixed vector 1f;o. Then at time t, according to Postulate 3,
the state of this system is represented by 1f;(t) = Ut1f;o, where
Ut = e-itH/h and H is the
energy operator. Consequently, the mean value of the observable a
at time
t is a(t) = (a.1f;(t), tI'(t» = (aUt tl'o ,Uttl'o). The operator Ut
being unitary, the
adjoint operator ofUt coincides with the inverse operator, (Ut)-l =
U- t. Hence
(3.1).
The distribution function of the observable a depends on time in a
similar
manner. In stating the basic concepts of quantum mechanics in §1 we
followed
the so-called Schrodinger picture (which we shall adhere to
henceforth): we
assumed time-dependence of the state vectors only and not of the
operators
of observables. One might assume instead (the Heisenberg picture)
that the
operators change in time and the state vectors remain invariant,
and if at time
t = 0 a certain observable a has a corresponding operator a, then
at time t
this observable must have the corresponding operator u_taUt .
Clearly, both
these approaches are equivalent, since they lead to the same
dependence of the
probability distribution of an arbitrary physical quantity on
time.
Proposition 3.1. If1f;o E DH, a.tI'(t) E DH for alit and the
operator a zs
bounded (or at least HtI'(t) E Do. and the vector-function atl'(t)
is continuous),
then the function a(t) is differentiable and the equality
(3.2)
holds.
a(s + t) - a(s) _ «aUt - uta)tI'(s) ,Uttl'(s» _ t t
= (a (Ut ; 1) tI'(s) ,Uttl'(s») _ (Ut ; 1) a.tI'(s) ,Uttl'(s»)
,
where I is the identity operator. Since it is supposed that atl'(s)
E DH, the
properties of the one-parameter group Ut (see S1.1.6 and the
analysis of the
16 CHAPTER 1
hence
lim(Ut - IO,t/J(s),Utt/J(s)) = ih(HO,t/J(s),t/J(s)). t-O t
When the operator a is unbounded, it is not clear whether there
exists a limit of the expression O,U\-I t/J(s) as t -4 O. However,
by the self-adjointness of a.,
( Ut - I ) (Ut - I )o'- t
-t/J(s), Utt/J(s) = -t-t/J(s), aUtt/J(s) ,
and the vector o'utt/J(s) = o't/J(t + s) is, by assumption,
continuously dependent on t. Therefore
lim (a. Ut - I t/J(s), Utt/J(s)) = ih(Ht/J(s),O,t/J(s)) =
ih(o'Ht/J(s), t/J(s)). t-O t
Hence we obtain (3.2) by setting t = s. •
If A, B are arbitrary linear operators, the operator AB - BA is
called the
commutator of A and B and is denoted by [A, B].
From (3.2) it is clear that if one can find an observable bsuch
that bt/J(t) = HH,u]t/J(t), then d~t) = bet) where bet) is the mean
value of b in the state
t/J(t). Therefore, the observable a is said to be time
differentiable if the operator
t(Ho' - a.H) has a unique self-adjoint extension b. The observable
b is called
then the time derivative of the observable a, and is denoted by ~~.
We usually
denote by the symbol HH,o'] not only the operator i(Ho' - aH) but
also its
self-adjoint extension. The observable ~~ is thus defined by the
equality:
dO, i dt = h[H,o']. (3.3)
In classical mechanics a physical quantity is represented by a
function
a(ql, ... ,qn, Pl, ... ,Pn, t) of generalized coordinates (ql,""
qn), generalized
momenta Pl, ... ,Pn, and time t. If the function a does not
explicitly depend
on time, that is, ~~ = 0, then, as is known, the total time
derivative of a is
given by the formula: da dt = [1£, a],
GENERAL CONCEPTS OF QUANTUM MECHANICS 17
where 1i is the Hamiltonian function expressing the energy in terms
of the coordinates and momenta, and
[1i, a] = ~ (01i oa _81i oa) (;t OPi Oqi Oqi OPi
is the so-called Poisson bracket of1i and a. Specifically, the
equations of motion of a mechanical system with the Hamiltonian 1i
have the form:
dqi 81i dPi o1i- = [1i,qi] =-j - = [1i,pd = --. dt OPi dt Oqi
Thus, the commutator of operators, multiplied by k, is the quantum
analogue of the Poisson bracket.
Proposition 3.2. An observable a is time-independent if and only if
the cor
responding operator a commutes with the energy operator.
Proof If a commutes with the energy operator H, then it also
commutes with
the evolution operators Ut = e- ttH. Consequently,
and the corresponding distribution function P",(A) is
time-independent.
Conversely, if for any '¢ the distribution function P",(A) is
time-indepen
dent, then (Ut-1E>.Ut,¢,,¢) = (E>.'¢,'¢) for any vector '¢ E
£. Then the opera
tors E>. and Ut commute for all t and Aj this means that the
operators il and
H also commute. In this case, of course, [H, il] = 0, da/dt = 0,
and the mean value a is time-independent. • Time-independent
observables are called constants of motion (as in clas
sical mechanics). Since the energy operator H commutes with itself,
it follows
from Proposition 3.3 that energy is an example of a constant of
motion (the
energy conservation law). Clearly, the energy conservation law may
fail, if
external forces act on a system. In this case, the above assumption
on the
time-independence of the energy operator H (in the Schrodinger
picture) may
not he true.
1.4. Quantization
Let us consider a classical dynamic system ~cl with n degrees of
freedom. * Its state is uniquely determined by the values of
generalized coordinates ql, ... , qn
* Here and in what follows, the non-relativistic theory is
considered.
18 CHAPTER 1
and generalized momenta Pl, ... ,Pn' Assume, for simplicity, that
the energy
1i of the system does not explicitly depend on time and is
expressed in terms
of momenta and coordinates,
(4.1)
where the mj are constants and each coordinate qj can range over
the real axis.
A typical example is a system of I particles (material points)
under poten
tial interaction. It has n = 31 degrees of freedom, and q3k-2,
q3k-l, q3k are
the Cartesian coordinates of the kth particle, P3k-2, P3k-1, P3k
the compo
nents of momentum, and m3k-2 =m3k-l =m3k the mass of the kth
particle,
V(ql' ... ,qn) the potential energy.
There exists a non-rigorous heuristic recipe for the construction
of a quan
tum mechanical system (!qm which is the quantum ana.logue of the
(!c\ system
and has the latter as its limit *. According to this recipe, the
state space I:- of the
system (!qm should be ta.ken as the space L 2(Rn ) of
complex-valued square
integrable functions of n real variables ql, ... , qn with respect
to Lebesgue
measure. Each coordinate qj should be associated with the position
operator
iij : (iij1/J)(ql"" ,qn) = qj1/J(ql"" ,qn); as with any
multiplication operator by a measurable real-valued function, the
operator qj is self-adjoint. Each mo
mentum Pk is associated with the self-adjoint operator Pk =
!ic.,ft-, where h is I UXk
Planck's constant and i the imaginary unit. Under the Fourier
transformation
this operator turns into the above position operator multiplied by
a constant.
The self-adjoint extension H of the differential operator **
(4.2)
is to be taken as the energy operator, usually referred to as the
Schrodinger
operator.
* In Chapter 5 we shall explain in what sense a quantum-mechanical
system may have the classical one as a limit. ** If the domain of a
differential operator is not specified, we assume that
this domain coincides with the set of all infinitely differentiable
functions with
compact support.
GENERAL CONCEPTS OF QUANTUM MECHANICS 19
dqk __1 (!:-~) dt - mk i {)qk .
Thus, Pk =mk~, that is, the relation between coordinates and
momenta is the same as in classical mechanics. Clearly, this
reasoning cannot be regarded
as quite rigorous, since the energy operator H was not well enough
defined.
It is easy to verify by direct calculation that the momentum and
position
operators satisfy the following relations:
The question of existence and uniqueness of this self-adjoint
extension re
quires special investigation in each particular case. Further on,
slightly abusing
the notation, we frequently denote both differential operators and
their self
adjoint extensions by the same symbol.
We differentiate the observable of a coordinate with respect to
time. By
(3.3) ~ = t[H, qk]; using formula (4.2) and performing a formal
calculation we easily obtain
[Pk,qk]=~I; [Pk,qj] =0, k:f:j, (4.3) t
where I is the identity operator. These relations (derived in 1925
by Born and
Jordan) are conventionally referred to as the Heisenberg
commutation relations.
In classical mechanics, momenta and coordinates are connected in a
similar way
by the Poisson bracket, so the analogy between the commutator and
the Poisson
bracket noted in §3 is confirmed once again.
The expression (4.2) of the energy operator H can be derived if the
vari
ables qj and Pj are formally replaced in formula (4.1) by the
operators qj
and 'Pi. This procedure is frequently expressed by the symbolic
formula H = 1-l (ql, ... ,qn,Pl, ... ,Pn). The quantum analogues of
some other important
physical quantities (for example, the angular momenta) can be
constructed in
exactly the same manner. Nevertheless, one cannot in this way
establish a
one-to-one correspondence between classical physical quantities
(that can be
expressed by arbitrary functions of momenta and coordinates) and
self-adjoint
operators in the state space £: since the product of the operators
Pj and qj
depends on the order of factors, ambiguity arises even in the
simplest cases.
By contrast, one can take a more or less arbitrary function of
commuting
self-adjoint operators (see S1.1.5). It is therefore easy to
specify the quantum
analogues for such observables that depend only on momenta or on
coordinates.
This transition from a classical mechanical system ~cl to a
quantum
mechanical one ~qm is called quantization. We assume that the
energy of the
20 CHAPTER 1
system <tel is expressed in terms ofthe coordinates and momenta
by (4.1). One
could have eliminated this restriction but it would of necessity
lead to much
more complicated quantization rules. In the general case, numerous
quantiza
tion techniques can be applied but there is essentially only one
way to construct
self-adjoint operators satisfying the Heisenberg relations and
natural additional
restrictions (see Exercise 8.7 below).
It should be borne in mind that there exist important
quantum-mechanical
systems unobtainable from a classical system by quantization. Such
are, pri
marily, particles with spin (electrons, protons, etc.) and systems
consisting of
identical particles (for example, many-electron systems).
The state of a system <tqm at the moment t is represented by the
function
t/J(ql, ... ,qn;t) which for each fixed t belongs to the space I:-
= L2(Rn). In this
case,
and the Schrodinger equation is
. eN h2 n 1 ePt/J zh-at = -2I: m' {) ? + V(ql,"" qn)t/J.
(4.4)
j=l J qJ
The function t/J(ql, ... ,qn; t) depending on n + 1 real variables
is called the wave function of the system <tqm . The function
t/J(ql, ... ,qn) of n variables
representing the state of the system <tqm at a fixed time is
also often called the
wave function.
It is known that in classical mechanics and in relativity theory it
is possible
to draw a far-reaching analogy between momentum and coordinate on
the one
hand, and the energy and time on the other. It is interesting that
this analogy
is also preserved in quantum mechanics: applying the momentum
operator to
a wave function we differentiate it (up to the factor ~) with
respect to the
corresponding coordinate and applying the energy operator to a wave
function,
as seen from the Schrodinger equation, we differentiate this wave
function with
respect to time (up to the factor ih = -~).
The operator E(k) of the spectral family corresponding to the
position
operator qk is the multiplication operator by the function that
takes the value
1 if qk :::; Aand the value 0 if qk > A. Therefore (see §1) in
the state t/J the joint
distribution function of the variables ql,"" qn has the following
form:
P1jJ(Al, ... ,An) = J:~ .. ·1: It/J(ql,"" qnWdql ... dqn.
GENERAL CONCEPTS OF QUANTUM MECHANICS 21
Consequently, the squared absolute value of the wave function
coincides with the
density of the joint distribution of the variables ql, ... , qn,
and the probability
for a measurement to find the values of the coordinates ql, ... ,
qn in the intervals
(al' bd, ... , (an, bn) respectively is equal to
This interpretation of the wave function and, correspondingly, the
statis
tical interpretation of quantum mechanics were proposed in 1926 by
M. Born.
The isomorphism of the state space {, of a quantum-mechanical
system
on a certain space of functions is named the representation of this
system.* Above we described the so-called position representation
of the system lEqm ;
this representation is not the only possible one; nor is it the
only one of interest.
Let us examine the operator
(4.5)
where q = (ql, ... ,qn), P=(Pl,." ,Pn), pq =I:.J=l Pjqj , dq = dql
dq2 ... dqn,
and the integral is taken over the whole space R n .
This operator is reduced to the Fourier-Plancherel transformation
by the
trivial substitution of variables: Pk = pdVh, qk = qk/Vh, k = 1,
... ,n.
Consequently, the operator F is an isometric isomorphism of the
space of
square-integrable functions of the variables ql, ... , qn onto the
space of square
integrable functions of the variables Pl, ... ,Pn. Thus a new
representation of
the system ~qm called the momentum representation naturally
arises.
A state of the system, given in the coordinate representation by a
func
tion t/J(ql, ... ,qn), is expressed in the momentum representation
by the function
J";(Pl> ... ,Pn) called sometimes the wave function in the
momentum representa
tion. From the known properties of the Fourier-Plancherel transform
it follows
that
( ) 1 J-( )i pqt/J q = (21rh )n/2 t/J P e"li dp, (4.6)
* In textbooks on quantum mechanics there are usually considered
represen tations of a special form, constructed with the help of a
system of commuting
self-adjoint operators having a simple joint spectrum. We discuss
such repre
sentations in §5.
22 CHAPTER 1
where the notations are similar to those used in formula (4.5). In
this case ih-a a Pk
corresponds in the momentum representation to the position operator
iik and the multiplication operator by the variable Pk corresponds
to the momentum
operator Pk. Therefore, 1¢(Pl, ... ,PnW is exactly the density of
the joint dis
tribution of the variables Pl , ... ,Pn, so that the probability of
finding by a mea surement that the values of momenta are in the
intervals (al,bl), ... ,(an,bn)
is equal to
Exercises
Let us denote by 2(n the linear space of all the analytical
functions j(z) = j(Zl' ... ,zn) of n complex variables Zl =Xl +
iYl, ... ,Zn =Xn + iYn such that
here and below we use the same notation as in formulas (4.5) and
(4.6), that
is, Z = (Zl, ... , zn), IzI2= L~=1IzkI2 = L~=l ZkZk, dx = dXl ...
dxn , dy = dYl ... dYn. We then define the scalar product in 2(n as
follows:
(4.7)
Finally, consider the integral operator on the space L2(Rn ) of
square
integrable functions of real variables ql, ... ,qn with respect to
Lebesgue mea-
sure:
U : ¢(q) --+ j(z) = ( I«z, q)¢(q)dq,Jan where
1. Prove that the integral operator U is a one-to-one mapping of
L2(Rn )
onto 2(n and that the inverse operator U- l is defined by
where the limit is understood in the sense of the metric of L 2(Rn
).
GENERAL CONCEPTS OF QUANTUM MECHANICS 23
2. Prove that mn is a complete Hilbert space with respect to the
scalar
product (4.7) and that U is an isomorphism between the Hilbert
spaces L 2(Rn )
and mn .
3. Prove that the functions Z:l ... z~n / ..jk1 ! ... kn ! with
non-negative inte
gers k1 , •.. ,kn form an orthonormal basis in the space mn .
Relate the expan
sion of elements of the space ~n with respect to this basis to the
Taylor series
expansion of analytical functions.
4. Let us identify the above space L2(Rn ) with the state space of
a
quantum-mechanical system (!qm in the coordinate representation.
The in
tegral operator U will then determine a new representation of the
system (!qm.
Prove that in this representation the form of the operator qk is
~(Zk + &~k)
and that of Pk is ~(a~k - Zk).
1.5. The Uncertainty Relations and Simultaneous Measurability
of
Physical Quantities
Consider arbitrary observables a and b whose corresponding
self-adjoint oper
ators are a and b. We fix a normalized vector 'I/J in the state
space such that (ab - ba)'I/J is defined. Let aand bbe the mean
values (that is the expectations) of a and b in the state 'I/J.
Indeterminacy of the results of measuring the observ abIes a and b
is characterized by the variances of the corresponding random
variables:
Lla = .;&t = lIa'I/J - a'I/Jll and Llb = V6b = IIb'I/J -
b'I/JlI·
Proposition 5.1. In the state 'I/J the mean-square deviations Lla
and Llb of the
observables a and b from their mean values satisfy the following
inequality:
Lla· Llb ~ ~I(ab - ba)'I/J, 'I/J)I· (5.1)
Proof Set 0.1 =a - a' I, b1 =b- b· I, where I is the identity
operator. It is
easy to verify that ab - ba = 0.161- b1a1. Therefore
I«ab - ba)'I/J, 'I/J)I = 1«a1b1- b1aI)'I/J, 'I/J)I = l(a1b1'I/J,
'I/J)-
-(b1a1'I/J,'I/J)1 = l(b1'I/J,a1'I/J) - (al'I/J,b1'I/J)1 =
24 CHAPTER 1
as required. •
Observables a and b are called canonically conjugate if the
relation iib-bii = hIli is true. In this case the right-hand side
of the inequality (5.1) does not
depend on 1/J and coincides with ~. Thus we have proved the
following
Proposition 5.2. If the observables a and b are canonically
conjugate, then
in any state the mean-square deviations ~a and ~b of these
observables from
their mean values satisfy the inequality
h ~a· ~b >-.- 2 (5.2)
For the systems described in §4 the Heisenberg commutation
relations (see
the equality (4.3)) show that a coordinate and the momentum
component cor
responding to it are canonically conjugate. The following statement
is therefore
true.
Corollary 5.1. For any fixed state, the uncertainties of the
results of measuring
the coordinate q" and the corresponding momentum component PIc
satisfy
(5.3)
The inequalities (5.2) and (5.3) constitute the exact forms of the
famous Heisen
berg uncertainty relations. In the case when Planck's constant
cannot be re
garded as negligible, the relation (5.3) shows that if a
microparticle is exactly
localized in space, no definite momentum can be assigned to it and
if a defi
nite momentum can be assigned (with a high degree of accuracy) to a
particle,
then, in this state, it is not localized in space. The existence of
relations of
type (5.3) was discovered in 1927 by W. Heisenberg when he studied
methods
of measuring coordinates and momenta of particles. For example, the
position
of a particle in space can be determined by illuminating it. In
this case photons
colliding with the particle tramsmit a part of their momenta to it
so that the
value of the momentum of the latter will have a certain
indeterminacy. The
GENERAL CONCEPTS OF QUANTUM MECHANICS 25
less the length of the light wave, the more exact is the
determination of the
particle's position. However, the energy and momentum of the photon
increase
with a decrease in the wave length and the uncertainty of the
particle's mo
mentum increases. A quantitative analysis of this result leads to
relations of
the (5.3) type. The statements proved above show that if quantum
mechanics
is true, other methods of measurement should also lead to a similar
result.
The uncertainty relations are closely associated with the question
as to
what physical quantities can be measured simultaneously. It is
believed that
the process of measuring an arbitrary observable a can be organized
so that
the result of measurement should be reproducible. This means that
if the first
measurement is followed rapidly enough (so that the state of the
system does
not have time to change in accordance with the Schrodinger
equation) by the
second measurement, then the result of this second measurement will
with cer
tainty be confined between some A1 and A2' These can be determined
from
the first measurement. * In this case, the interval (AI, A2) can
(by organizing
the measurement process) be made arbitrarily small. Thus, by
organizing the
measurement of the observable a in suitable fashion the system can
be trans
ferred into a state with .6.a however small. Therefore, if
observables a and bare
simultaneously measurable, then there exists a state in which .6.a
and .6.b are
arbitrarily small. From this, as well as from the inequality (5.2),
it follows that
canonically conjugate observables cannot be simultaneously
measured; such
measurements require mutually excluding experimental set-ups.
In §1 a more general statement was postulated: observablesa1' ... ,
an
are simultaneously measurable if and only if the operators aI, ...
,an commute. Proposition 5.1 can be regarded as an argument in
favour of this postulate. A
detailed explanation why commuting operators must correspond to
simultane
ously measurable observables can be found in the book by von
Neumann [1].
If operators a1,'" ,an commute, then (see 81.1.5) there exists a
self-adjoint operator a and functions ft, ... ,fn such that a1
=11(a), ... ,an = In(a). By Postulate 4 (see §1), the operator
acorresponds to a certain observable a. Be-
* It should be noted, however, that some of the measurement methods
used
in experiments do not ensure the reproducibility of results.
Moreover, when
performing measurements we usually deal not with one micf(~·object
but with a
"statistical ensemble" of them (see von Neumann [1], Mandelshtamm
[1], Fock [1], Kholevo [1]).
26 CHAPTER 1
cause of this, measurements of the observable a also yield the
values of all the
al, ... , an, so that it is natural to regard these observables as
simultaneously
measurable. The aforesaid is also generalized to the case when
there exist
superselection rules, specified by a set of mutually orthogonal
(that is, com
muting and self-adjoint) projections Pa . In this case the
operators al, ... ,an
commute with all the Pa , and the operator acan be chosen so that
Pa = fa(a)
for a certain function fa, so that a commutes with all the Pa and,
therefore,
corresponds to an observable.
If the observables al, ... ,an are simultaneously measurable and if
every
observable simultaneously measurable with al, ... , an is a
function of these
observables then the observables aI, ... ,an are said to form a
complete set. *
Proposition 5.3. The observables al, ... , an form a complete set
if and only
if the operators aI, ... ,an commute and have a simple joint
spectrum. In this
case there exists an isomorphism of the state space J:, onto the
space L2(Rn , dj.l)
of all the functions of n variables AI, ... ,An, square-integrable
with respect to
a certain measure dj.l, such that ai turns into multiplication
operator by an
independent variable Ai (the measure dj.l is not unique).
This statement follows from the results formulated in Supplement 1
(Pro
position S1.10, Theorem S1.4, Exercises S1.8.3 and S1.8.4).
Proposition 5.3
shows that a complete set of observables can be related to a
certain represen
tation of quantum-mechanical system. Such representations are
exemplified
by the position and momentum representations discussed in §4. In
the former
case the complete set consists of coordinates, and in the latter,
of momenta.
In classical physics, in order to determine completely the state of
a system,
one has to know the values of all coordinates and momenta (at a
fixed moment
of time). For a quantum-mechanical system a complete set obviously
cannot
include both the coordinates and the momenta.
It is possible in principle to determine the state of a system from
the result
of measuring a complete set of observables. Let us consider as an
example an
observable a for which the operator a has a simple pUJ'e point
spectrum, that is, there exists in the state space an orthonormal
basis 1/JI, ... ,1/Jn, ... for which
a1/Ji = Ai.,pi and Ai :f Aj for i :f j. In this case the complete
set is reduced to a single observable a. Let us assume that the
results of mea