the schr¶dinger equation

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 £.
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.
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).
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
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
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.
(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],
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.
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.
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 ).
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
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

the schr¶dinger equation

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