Mathematical concepts of quantum mechanics (2nd edition

Universitext

www.pdfgrip.com


Universitext
Series Editors:
Sheldon Axler
San Francisco State University
Vincenzo Capasso
Universita` degli Studi di Milano
Carles Casacuberta
Universitat de Barcelona
Angus J. MacIntyre
Queen Mary, University of London
Kenneth Ribet
University of California, Berkeley
Claude Sabbah

CNRS, Ecole
Polytechnique
Endre Săuli
University of Oxford
Wojbor A. Woyczynski
Case Western Reserve University

Universitext is a series of textbooks that presents material from a wide variety of
mathematical disciplines at master’s level and beyond. The books, often well
class-tested by their author, may have an informal, personal even experimental
approach to their subject matter. Some of the most successful and established books

in the series have evolved through several editions, always following the evolution
of teaching curricula, to very polished texts.
Thus as research topics trickle down into graduate-level teaching, first textbooks
written for new, cutting-edge courses may make their way into Universitext.

For further volumes:
www.springer.com/series/223

www.pdfgrip.com


Stephen J. Gustafson
Israel Michael Sigal

Mathematical Concepts
of Quantum Mechanics
Second Edition

123
www.pdfgrip.com


Stephen J. Gustafson
University of British Columbia
Dept. Mathematics
Vancouver, BC V6T 1Z2
Canada


Israel Michael Sigal

University of Toronto
Dept. Mathematics
40 St. George Street
Toronto, ON M5S 2E4
Canada


ISBN 978-3-642-21865-1
e-ISBN 978-3-642-21866-8
DOI 10.1007/978-3-642-21866-8
Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2011935880
Mathematics Subject Classification (2010): 81S, 47A, 46N50
c Springer-Verlag Berlin Heidelberg 2011
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,
reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication
or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,
1965, in its current version, and permission for use must always be obtained from Springer. Violations are
liable to prosecution under the German Copyright Law.
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,
even in the absence of a specific statement, that such names are exempt from the relevant protective laws
and regulations and therefore free for general use.
Cover design: deblik, Berlin
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)

www.pdfgrip.com

Preface

Preface to the second edition
Oneof the main goals motivating this new edition was to enhance the
elementary material. To this end, in addition to some rewriting and reorganization, several new sections have been added (covering, for example, spin, and
conservation laws), resulting in a fairly complete coverage of elementary topics.
A second main goal was to address the key physical issues of stability of
atoms and molecules, and mean-field approximations of large particle systems.
This is reflected in new chapters covering the existence of atoms and molecules,
mean-field theory, and second quantization.
Our final goal was to update the advanced material with a view toward
reflecting current developments, and this led to a complete revision and reorganization of the material on the theory of radiation (non-relativistic quantum
electrodynamics), as well as the addition of a new chapter.
In this edition we have also added a number of proofs, which were omitted
in the previous editions. As a result, this book could be used for senior level
undergraduate, as well as graduate, courses in both mathematics and physics
departments.
Prerequisites for this book are introductory real analysis (notions of vector space, scalar product, norm, convergence, Fourier transform) and complex analysis, the theory of Lebesgue integration, and elementary differential
equations. These topics are typically covered by the third year in mathematics
departments. The first and third topics are also familiar to physics undergraduates. However, even in dealing with mathematics students we have found it
useful, if not necessary, to review these notions, as needed for the course.
Hence, to make the book relatively self-contained, we briefly cover these subjects, with the exception of Lebesgue integration. Those unfamiliar with the
latter can think about Lebesgue integrals as if they were Riemann integrals.
This said, the pace of the book is not a leisurely one and requires, at least for
beginners, some amount of work.
Though, as in the previous two issues of the book, we tried to increase
the complexity of the material gradually, we were not always successful, and

www.pdfgrip.com

VI

Preface

first in Chapter 12, and then in Chapter 18, and especially in Chapter 19,
there is a leap in the level of sophistication required from the reader. One
may say the book proceeds at three levels. The first one, covering Chapters 111, is elementary and could be used for senior level undergraduate, as well as
graduate, courses in both physics and mathematics departments; the second
one, covering Chapters 12 - 17, is intermediate; and the last one, covering
Chapters 18 - 22, advanced.
During the last few years since the enlarged second printing of this book,
there have appeared four books on Quantum Mechanics directed at mathematicians:
F. Strocchi, An Introduction to the Mathematical Structure of Quantum Mechanics: a Short Course for Mathematicians. World Scientific, 2005.
L. Takhtajan, Quantum Mechanics for Mathematicians. AMS, 2008.
L.D. Faddeev, O.A. Yakubovskii, Lectures on Quantum Mechanics for Mathematics Students. With an appendix by Leon Takhtajan. AMS, 2009.
J. Dimock, Quantum Mechanics and Quantum Field Theory. Cambridge Univ.
Press, 2011.
These elegant and valuable texts have considerably different aims and rather
limited overlap with the present book. In fact, they complement it nicely.
Acknowledgment: The authors are grateful to I. Anapolitanos, Th. Chen, J.
Faupin, Z. Gang, G.-M. Graf, M. Griesemer, L. Jonsson, M. Merkli, M. Mă
uck,
Yu. Ovchinnikov, A. Soer, F. Ting, T. Tzaneteas, and especially J. Fră
ohlich,
W. Hunziker and V. Buslaev for useful discussions, and to J. Feldman, G.-M.
Graf, I. Herbst, L. Jonsson, E. Lieb, B. Simon and F. Ting for reading parts
of the manuscript and making useful remarks.
Vancouver/Toronto,
May 2011

Stephen Gustafson
Israel Michael Sigal

Preface to the enlarged second printing
For the second printing, we corrected a few misprints and inaccuracies; for
some help with this, we are indebted to B. Nachtergaele. We have also added
a small amount of new material. In particular, Chapter 11, on perturbation
theory via the Feshbach method, is new, as are the short sub-sections 13.1
and 13.2 concerning the Hartree approximation and Bose-Einstein condensation. We also note a change in terminology, from “point” and “continuous”
spectrum, to the mathematically more standard “discrete” and “essential”
spectrum, starting in Chapter 6.
Vancouver/Toronto,
July 2005

Stephen Gustafson
Israel Michael Sigal

www.pdfgrip.com


Preface

VII

From the preface to the first edition
The first fifteen chapters of these lectures (omitting four to six chapters
each year) cover a one term course taken by a mixed group of senior undergraduate and junior graduate students specializing either in mathematics or
physics. Typically, the mathematics students have some background in advanced analysis, while the physics students have had introductory quantum
mechanics. To satisfy such a disparate audience, we decided to select material

which is interesting from the viewpoint of modern theoretical physics, and
which illustrates an interplay of ideas from various fields of mathematics such
as operator theory, probability, differential equations, and differential geometry. Given our time constraint, we have often pursued mathematical content
at the expense of rigor. However, wherever we have sacrificed the latter, we
have tried to explain whether the result is an established fact, or, mathematically speaking, a conjecture, and in the former case, how a given argument
can be made rigorous. The present book retains these features.

Vancouver/Toronto,
Sept. 2002

Stephen Gustafson
Israel Michael Sigal

www.pdfgrip.com


•

www.pdfgrip.com


Contents

1

Physical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
The Double-Slit Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3
State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
The Schrăodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1
Conservation of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2
Self-adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3
Existence of Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4
The Free Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3

Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
Mean Values and the Momentum Operator . . . . . . . . . . . . . . . .
3.2
Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
The Heisenberg Representation . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5.1
Probability current . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19
19
20
21
22
23
24

4

Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Quantization and Correspondence Principle . . . . . . . . . . . . . . . .
4.3
Complex Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
Supplement: Hamiltonian Formulation of Classical
Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27
27
29
32

5

Uncertainty Principle and Stability of Atoms and
Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
The Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . . . . . . .
5.2
A Refined Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
Application: Stability of Atoms and Molecules . . . . . . . . . . . . .

www.pdfgrip.com

1
2
4
5
5

36
41
41
42
43


X

Contents

6

Spectrum and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1
The Spectrum of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
Bound and Decaying States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3
Spectra of Schră
odinger Operators . . . . . . . . . . . . . . . . . . . . . . . . .
6.4
Supplement: Particle in a Periodic Potential . . . . . . . . . . . . . . .

47
47
50
53
57

7

Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
The Infinite Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
The Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3
The Square Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4
A Particle on a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5

The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6
The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7
A Particle in a Constant Magnetic Field . . . . . . . . . . . . . . . . . .
7.8
Linearized Ginzburg-Landau Equations
of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61
61
62
62
63
64
66
69

8

Bound States and Variational Principle . . . . . . . . . . . . . . . . . . .
8.1
Variational Characterization of Eigenvalues . . . . . . . . . . . . . . . .
8.2
Exponential Decay of Bound States . . . . . . . . . . . . . . . . . . . . . . .
8.3
Number of Bound States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75
75

81
81

9

Scattering States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1
Short-range Interactions: μ > 1 . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2
Long-range Interactions: μ ≤ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3
Wave Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4
Appendix: The Potential Step and Square Well . . . . . . . . . . . .

89
90
92
93
96

70

10 Existence of Atoms and Molecules . . . . . . . . . . . . . . . . . . . . . . . . 99
10.1 Essential Spectra of Atoms and Molecules . . . . . . . . . . . . . . . . . 99
10.2 Bound States of Atoms and BO Molecules . . . . . . . . . . . . . . . . . 101
10.3 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
11 Perturbation Theory: Feshbach-Schur Method . . . . . . . . . . . . 107
11.1 The Feshbach-Schur Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
11.2 Example: the Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

11.3 Example: Time-Dependent Perturbations . . . . . . . . . . . . . . . . . . 114
11.4 Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . . . . . 119
11.5 Appendix: Projecting-out Procedure . . . . . . . . . . . . . . . . . . . . . . 122
11.6 Appendix: Proof of Theorem 11.1 . . . . . . . . . . . . . . . . . . . . . . . . 123
12 General Theory of Many-particle Systems . . . . . . . . . . . . . . . . . 127
12.1 Many-particle Schrăodinger Operators . . . . . . . . . . . . . . . . . . . . . 127
12.2 Separation of the Centre-of-mass Motion . . . . . . . . . . . . . . . . . . 129
12.3 Break-ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

www.pdfgrip.com


Contents

12.4
12.5
12.6
12.7
12.8

XI

The HVZ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Intra- vs. Inter-cluster Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Exponential Decay of Bound States . . . . . . . . . . . . . . . . . . . . . . . 136
Remarks on Discrete Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Scattering States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

13 Self-consistent Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
13.1 Hartree, Hartree - Fock and Gross-Pitaevski equations . . . . . . 141

13.2 Appendix: BEC at T=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
14 The
14.1
14.2
14.3

Feynman Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
The Feynman Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Generalizations of the Path Integral . . . . . . . . . . . . . . . . . . . . . . 152
Mathematical Supplement: the Trotter Product Formula . . . . 153

15 Quasi-classical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
15.1 Quasi-classical Asymptotics of the Propagator . . . . . . . . . . . . . 156
15.2 Quasi-classical Asymptotics of Green’s Function . . . . . . . . . . . . 160
15.2.1 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
15.3 Bohr-Sommerfeld Semi-classical Quantization . . . . . . . . . . . . . . 163
15.4 Quasi-classical Asymptotics for the Ground State Energy . . . . 165
15.5 Mathematical Supplement: The Action
of the Critical Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
15.6 Appendix: Connection to Geodesics . . . . . . . . . . . . . . . . . . . . . . . 170
16 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
16.1 Complex Deformation and Resonances . . . . . . . . . . . . . . . . . . . . 173
16.2 Tunneling and Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
16.3 The Free Resonance Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
16.4 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
16.5 Positive Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
16.6 Pre-exponential Factor for the Bounce . . . . . . . . . . . . . . . . . . . . 184
16.7 Contribution of the Zero-mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
16.8 Bohr-Sommerfeld Quantization for Resonances . . . . . . . . . . . . . 186
17 Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

17.1 Information Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
17.2 Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
17.3 Quantum Statistics: General Framework . . . . . . . . . . . . . . . . . . 197
17.4 Hilbert Space Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
17.5 Quasi-classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
17.6 Reduced Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
17.7 Irreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

www.pdfgrip.com


XII

Contents

18 The
18.1
18.2
18.3
18.4
18.5
18.6
18.7

Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Fock Space and Creation and Annihilation Operators . . . . . . . 209
Many-body Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Evolution of Quantum Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Relation to Quantum Harmonic Oscillator . . . . . . . . . . . . . . . . . 215
Scalar Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Mean Field Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Appendix: the Ideal Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
18.7.1 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . 223

19 Quantum Electro-Magnetic Field - Photons . . . . . . . . . . . . . . . 227
19.1 Klein-Gordon Classical Field Theory . . . . . . . . . . . . . . . . . . . . . . 227
19.1.1 Principle of minimum action . . . . . . . . . . . . . . . . . . . . . 227
19.1.2 Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
19.1.3 Hamiltonian System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
19.1.4 Complexification of the Klein-Gordon Equation . . . . 231
19.2 Quantization of the Klein-Gordon Equation . . . . . . . . . . . . . . . 232
19.3 The Gaussian Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
19.4 Wick Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
19.5 Fock Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
19.6 Quantization of Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . 241
20 Standard Model of Non-relativistic Matter
and Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
20.1 Classical Particle System Interacting with an
Electro-magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
20.2 Quantum Hamiltonian of Non-relativistic QED . . . . . . . . . . . . . 250
20.2.1 Translation invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
20.2.2 Fiber decomposition with respect
to total momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
20.3 Rescaling and Decoupling Scalar and Vector Potentials . . . . . . 255
20.3.1 Self-adjointness of H(ε) . . . . . . . . . . . . . . . . . . . . . . . . . 256
20.4 Mass Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
20.5 Appendix: Relative bound on I(ε) and Pull-through
Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
21 Theory of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
21.1 Spectrum of Uncoupled System . . . . . . . . . . . . . . . . . . . . . . . . . . 264

21.2 Complex Deformations and Resonances . . . . . . . . . . . . . . . . . . . 265
21.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
21.4 Idea of the Proof of Theorem 21.1 . . . . . . . . . . . . . . . . . . . . . . . . 269
21.5 Generalized Pauli-Fierz Transformation . . . . . . . . . . . . . . . . . . . 270
21.6 Elimination of Particle and High Photon Energy
Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
21.7 The Hamiltonian H0 (ε, z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
21.8 Estimates on the Operator H0 (ε, z) . . . . . . . . . . . . . . . . . . . . . . . 277

www.pdfgrip.com


Contents

XIII

21.9 Ground State of H(ε) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
21.10 Appendix: Estimates on Iε and HP¯ρ (ε) . . . . . . . . . . . . . . . . . . . . 280
21.11 Appendix: Key Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
22 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
22.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
22.2 A Banach Space of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
22.3 The Decimation Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
22.4 The Renormalization Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
22.5 Dynamics of RG and Spectra of Hamiltonians . . . . . . . . . . . . . . 293
22.6 Supplement: Group Property of Rρ . . . . . . . . . . . . . . . . . . . . . . . 298
23 Mathematical Supplement: Spectral Analysis . . . . . . . . . . . . . 299
23.1 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
23.2 Operators on Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
23.3 Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

23.4 Inverses and their Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
23.5 Self-adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
23.6 Exponential of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
23.7 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
23.8 The Spectrum of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
23.9 Functions of Operators and the Spectral
Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
23.10 Weyl Sequences and Weyl Spectrum . . . . . . . . . . . . . . . . . . . . . . 321
23.11 The Trace, and Trace Class Operators . . . . . . . . . . . . . . . . . . . . 327
23.12 Operator Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
23.13 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
23.14 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
24 Mathematical Supplement: The Calculus of Variations . . . . 339
24.1 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
24.2 The First Variation and Critical Points . . . . . . . . . . . . . . . . . . . 341
24.3 The Second Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
24.4 Conjugate Points and Jacobi Fields . . . . . . . . . . . . . . . . . . . . . . . 346
24.5 Constrained Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . 350
24.6 Legendre Transform and Poisson Bracket . . . . . . . . . . . . . . . . . . 351
24.7 Complex Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
24.8 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
25 Comments on Literature, and Further Reading . . . . . . . . . . . 359
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

www.pdfgrip.com


1
Physical Background

The starting point of quantum mechanics was Planck’s idea that electromagnetic radiation is emitted and absorbed in discrete amounts – quanta. Einstein
ventured further by suggesting that the electro-magnetic radiation itself consists of particles, which were then named photons. These were the first quantum particles and the first glimpse of wave-particle duality. Then came Bohr’s
model of an atom, with electrons moving on fixed orbits and jumping from orbit to orbit without going through intermediate states. This culminated first in
Heisenberg and then in Schră
odinger quantum mechanics, with the next stage
incorporating quantum electro-magnetic radiation accomplished by Jordan,
Pauli, Heisenberg, Born, Dirac and Fermi.
To complete this thumbnail sketch we mention two dramatic experiments.
The first one was conducted by E. Rutherford in 1911, and it established the
planetary model of an atom with practically all its weight concentrated in
a tiny nucleus (10−13 − 10−12 cm) at the center and with electrons orbiting
around it. The electrons are attracted to the nucleus and repelled by each
other via the Coulomb forces. The size of an atom, i.e. the size of electron
orbits, is about 10−8 cm. The problem is that in classical physics this model
is unstable.
The second experiment is the scattering of electrons on a crystal conducted
by Davisson and Germer (1927), G.P. Thomson (1928) and Rupp (1928), after
the advent of quantum mechanics. This experiment is similar to Young’s 1805
experiment confirming the wave nature of light. It can be abstracted as the
double-slit experiment described below. It displays an interference pattern for
electrons, similar to that of waves.
In this introductory chapter, we present a very brief overview of the basic
structure of quantum mechanics, and touch on the physical motivation for
the theory. A detailed mathematical discussion of quantum mechanics is the
focus of the subsequent chapters.

www.pdfgrip.com

2

1 Physical Background

1.1 The Double-Slit Experiment
Suppose a stream of electrons is fired at a shield in which two narrow slits
have been cut (see Fig. 1.1.) On the other side of the shield is a detector
screen.

shield
electron

slits

gun
screen
Fig. 1.1. Experimental set-up.
Each electron that passes through the shield hits the detector screen at
some point, and these points of contact are recorded. Pictured in Fig. 1.2 and
Fig. 1.3 are the intensity distributions observed on the screen when either of
the slits is blocked.

P1 (brightness)

Fig.1.2. First slit blocked.

www.pdfgrip.com


1.1 The Double-Slit Experiment

3

P2

Fig.1.3. Second slit blocked.
When both slits are open, the observed intensity distribution is shown in
Fig. 1.4.

P = P1 + P 2

Fig.1.4. Both slits open.
Remarkably, this is not the sum of the previous two distributions; i.e.,
P = P1 + P2 . We make some observations based on this experiment.
1. We cannot predict exactly where a given electron will hit the screen, we
can only determine the distribution of locations.
2. The intensity pattern (called an interference pattern) we observe when
both slits are open is similar to the pattern we see when a wave propagates
through the slits: the intensity observed when waves E1 and E2 (the waves
here are represented by complex numbers encoding the amplitude and

www.pdfgrip.com


4

1 Physical Background

phase) originating at each slit are combined is proportional to |E1 +E2 |2 =
|E1 |2 + |E2 |2 (see Fig. 1.5).

Fig.1.5. Wave interference.
We can draw some conclusions based on these observations.
1. Matter behaves in a random way.
2. Matter exhibits wave-like properties.
In other words, the behaviour of individual electrons is intrinsically random,
and this randomness propagates according to laws of wave mechanics. These
observations form a central part of the paradigm shift introduced by the theory
of quantum mechanics.

1.2 Wave Functions
In quantum mechanics, the state of a particle is described by a complex-valued
function of position and time, ψ(x, t), x ∈ R3 , t ∈ R. This is called a wave
function (or state vector). Here Rd denotes d-dimensional Euclidean space,
R = R1 , and a vector x ∈ Rd can be written in coordinates as x = (x1 , . . . , xd )
with xj ∈ R.
In light of the above discussion, the wave function should have the following
properties.
1. |ψ(·, t)|2 is the probability distribution for the particle’s position. That
is, the probability that a particle is in the region Ω ⊂ R3 at time t is
|ψ(x, t)|2 dx. Thus we require the normalization R3 |ψ(x, t)|2 dx = 1.
Ω
2. ψ satisfies some sort of wave equation.
For example, in the double-slit experiment, if ψ1 gives the state beyond the
shield with the first slit closed, and ψ2 gives the state beyond the shield with

www.pdfgrip.com


1.4 The Schră

odinger Equation

5

the second slit closed, then = ψ1 + ψ2 describes the state with both slits
open. The interference pattern observed in the latter case reflects the fact that
|ψ|2 = |ψ1 |2 + |ψ2 |2 .

1.3 State Space
The space of all possible states of the particle at a given time is called the state
space. For us, the state space of a particle will usually be the square-integrable
functions:
|ψ(x)|2 dx < ∞}
L2 (R3 ) := {ψ : R3 → C |
R3

(we can impose the normalization condition as needed). This is a vector space,
and has an inner-product given by
ψ, φ :=

R3

¯
ψ(x)φ(x)dx.

In fact, it is a Hilbert space (see Section 23.2 for precise definitions and mathematical details).

1.4 The Schră
odinger Equation
We now give a motivation for the equation which governs the evolution of

a particles wave function. This is the celebrated Schră
odinger equation. An
evolving state at time t is denoted by ψ(x, t), with the notation ψ(t)(x) ≡
ψ(x, t).
Our equation should satisfy certain physically sensible properties.
1. Causality: The state ψ(t0 ) at time t = t0 should determine the state ψ(t)
for all later times t > t0 .
2. Superposition principle: If ψ(t) and φ(t) are evolutions of states, then
αψ(t) + βφ(t) (α, β constants) should also describe the evolution of a
state.
3. Correspondence principle: In “everyday situations,” quantum mechanics
should be close to the classical mechanics we are used to.
The first requirement means that ψ should satisfy an equation which is firstorder in time, namely
∂
ψ = Aψ
(1.1)
∂t
for some operator A, acting on the state space. The second requirement implies
that A must be a linear operator.
We use the third requirement – the correspondence principle – in order
to find the correct form of A. Here we are guided by an analogy with the
transition from wave optics to geometrical optics.

www.pdfgrip.com


6

1 Physical Background

Wave Optics

→ Geometrical Optics

Quantum Mechanics → Classical Mechanics
In everyday experience we see light propagating along straight lines in accordance with the laws of geometrical optics, i.e., along the characteristics of the
equation
∂φ
= ±c|∇x φ|
(c = speed of light),
(1.2)
∂t
known as the eikonal equation. On the other hand we know that light, like
electro-magnetic radiation in general, obeys Maxwell’s equations which can
be reduced to the wave equation (say, for the electric field in the complex
representation)
∂2u
= c2 Δu,
(1.3)
∂t2
3

2
where Δ =
j=1 ∂j is the Laplace operator, or the Laplacian (in spatial
dimension three).
The eikonal equation appears as a high frequency limit of the wave equation when the wave length is much smaller than the typical size of objects.
iφ
Namely we set u = ae λ , where a and φ are real and O(1) and λ > 0 is the
ratio of the typical wave length to the typical size of objects. The real function

φ is called the eikonal. Substitute this into (1.3) to obtain

a
ă + 2i1 a 2 a 2 + i1 aă
= c2 (Δa + 2iλ−1 ∇a · ∇φ − λ−2 a|∇φ|2 + λ−1 aΔφ)
(where dots denote derivatives with respect to t). In the short wave approximation, λ
1 (with derivatives of a and φ O(1)), we obtain
−aφ˙ 2 = −c2 a|∇φ|2 ,
which is equivalent to the eikonal equation (1.2).
An equation in classical mechanics analogous to the eikonal equation is
the Hamilton-Jacobi equation
∂S
= −h(x, ∇S),
∂t

(1.4)

where h(x, k) is the classical Hamiltonian function, which for a particle of mass
1
|k|2 + V (x), and S(x, t)
m moving in a potential V is given by h(x, k) = 2m
is the classical action. We would like to find an evolution equation which
would lead to the Hamilton-Jacobi equation in the way the wave equation
led to the eikonal one. We look for a solution to equation (1.1) in the form
ψ(x, t) = a(x, t)eiS(x,t)/ , where S(x, t) satisfies the Hamilton-Jacobi equation

www.pdfgrip.com


1.4 The Schră

odinger Equation

7

(1.4) and is a parameter with the dimensions of action, small compared to a
typical classical action for the system in question. Assuming a is independent
of , it is easy to show that, to leading order, ψ then satisfies the equation
i

2
∂
ψ(x, t) = −
Δx ψ(x, t) + V (x)ψ(x, t).
∂t
2m

(1.5)

This equation is of the desired form (1.1). In fact it is the correct equation, and
is called the Schră
odinger equation. The small constant is Planck’s constant;
it is one of the fundamental constants in nature. For the record, its value is
roughly
≈ 6.6255 × 10−27 erg sec.
The equation (1.5) can be written as
i

∂
ψ = Hψ
∂t

(1.6)

where the linear operator H, called a Schră
odinger operator, is given by
H :=

2

2m

+ V .

Example 1.1 Here are just a few examples of potentials.
1.
2.
3.
4.
5.

Free motion : V ≡ 0.
A wall: V ≡ 0 on one side, V ≡ ∞ on the other (meaning ψ ≡ 0 here).
The double-slit experiment: V ≡ ∞ on the shield, and V ≡ 0 elsewhere.
The Coulomb potential : V (x) = −α/|x| (describes a hydrogen atom).
2
2
The harmonic oscillator : V (x) = mω
2 |x| .

We will analyze some of these examples, and others, in subsequent chapters.

www.pdfgrip.com


2
Dynamics

The purpose of this chapter is to investigate the existence and a key property
– conservation of probability – of solutions of the Schrăodinger equation for
a particle of mass m in a potential V . The relevant background material on
linear operators is reviewed in the Mathematical Supplement Chapter 23.
We recall that the Schră
odinger equation,
i


= H
t

(2.1)

2

where the linear operator H = 2m + V is the corresponding Schră
odinger
operator, determines the evolution of the particle state (the wave function),
ψ. We supplement equation (2.1) with the initial condition
ψ|t=0 = ψ0

(2.2)

where ψ0 ∈ L2 (R3 ). The problem of solving (2.1)- (2.2) is called an initial
value problem or a Cauchy problem.
Both the existence and the conservation of probability do not depend on
the particular form of the operator H, but rather follow from a basic property
– self-adjointness. This property is rather subtle, so for the moment we just
mention that self-adjointness is a strengthening of a much simpler property –
symmetry. A linear operator A acting on a Hilbert space H is symmetric if
for any two vectors in the domain of A, ψ, φ ∈ D(A),
Aψ, φ = ψ, Aφ .

2.1 Conservation of Probability
Since we interpret |ψ(x, t)|2 at a given instant in time as a probability distribution, we should have

www.pdfgrip.com


10

2 Dynamics

R3

|ψ(x, t)|2 dx ≡ 1

=
R3

(2.3)

|ψ(x, 0)|2 dx

at all times, t. If (2.3) holds, we say that probability is conserved.
Theorem 2.1 Solutions ψ(t) of (2.1) with ψ(t) ∈ D(H) conserve probability
if and only if H is symmetric.
Proof. Suppose ψ(t) ∈ D(H) solves the Cauchy problem (2.1)-(2.2). We compute
d
˙ ψ + ψ, ψ˙ = 1 Hψ, ψ + ψ, 1 Hψ
ψ, ψ = ψ,
dt
i
i
1
= [ ψ, Hψ − Hψ, ψ ]
i
(here, and often below, we use the notation ψ˙ to denote ∂ψ/∂t). If H is
symmetric then this time derivative is zero, and hence probability is conserved.
Conversely, if probability is conserved for all such solutions, then Hφ, φ =
φ, Hφ for all φ ∈ D(H) (since we may choose ψ0 = φ). This, in turn,
implies H is a symmetric operator. The latter fact follows from a version of
the polarization identity,
ψ, φ =

1
( φ+ψ
4

2

− φ−ψ

2

− i φ + iψ

2

+ i φ − iψ 2 ),

(2.4)

whose proof is left as an exercise below.
Problem 2.2 Prove (2.4).
Problem 2.3 Show that the following operators on L2 (R3 ) (with their natural domains) are symmetric:
xj (that is, multiplication by xj );
pj := −i ∂xj ;
2
H0 := − 2m Δ;
3
for f : R → R bounded, f (x) (multiplication operator) and f (p) :=
F −1 f (k)F (here F denotes Fourier transform);
5. integral operators Kf (x) = K(x, y)f (y) dy with K(x, y) = K(y, x) and,
say, K ∈ L2 (R3 × R3 ).
1.
2.
3.
4.

2.2 Self-adjointness
As was mentioned above the key property of the Schr´

odinger operator H
which guarantees existence of dynamics is its self-adjointness. We define this
notion here. More detail can be found in Section 23.5 of the mathematical
supplement.

www.pdfgrip.com


2.2 Self-adjointness

11

Definition 2.4 A linear operator A acting on a Hilbert space H is self-adjoint
if A is symmetric and Ran(A ± i1) = H.
Note that the condition Ran(A ± i1) = H is equivalent to the fact that the
equations
(A ± i)ψ = f
(2.5)
have solutions for all f ∈ H. The definition above differs from the one commonly used (see Section 23.5 of the Mathematical Supplement and e.g. [RSI]),
but is equivalent to it. This definition isolates the property one really needs
and avoids long proofs which are not relevant to us.
Example 2.5 The operators in Problem 2.3 are all self-adjoint.
Proof. We show this for p = −i ∂x on the space L2 (R). This operator is
symmetric, so we compute Ran(−i ∂x + i). That is, we solve
(−i ∂x + i)ψ = f,
which, using the Fourier transform (see Section 23.14), is equivalent to (k +
ˆ
i)ψ(k)
= fˆ(k), and therefore
ψ(x) = (2π )−1/2

R

eikx/

fˆ(k)
dk.
k+i

Now for any such f ∈ L2 (R),
ˆ
= |fˆ(k)| ∈ L2 (R),
(1 + |k|2 )1/2 |ψ(k)|
so ψ lies in the Sobolev space of order one, H 1 (R) = D(−i ∂x ), and therefore
Ran(−i ∂x + i1) = L2 . Similarly Ran(−i ∂x − i1) = L2 .
Problem 2.6 Show that xj , f (x) and f (p), for f real and bounded, and Δ
are all self-adjoint on L2 (R3 ) (with their natural domains).
In what follows we omit the identity operator 1 in expressions like A − z1.
The next result establishes the self-adjointness of Schrăodinger operators.
2

Theorem 2.7 Assume that V is real and bounded. Then H := − 2m Δ+V (x),
with D(H) = D(Δ), is self-adjoint on L2 (R3 ).
Proof. It is easy to see (just as in Problem 2.3) that H is symmetric. To
prove Ran(H ± i) = H, we will use the following facts proved in Sections 23.4
and 23.5 of the mathematical supplement:
1. If an operator K is bounded and satisfies K < 1, then the operator
1 + K has a bounded inverse.
2. If A is symmetric and Ran(A − z) = H for some z, with Im z > 0, then it
is true for every z with Im z > 0. The same is true for Im z < 0.

www.pdfgrip.com


12

2 Dynamics

3. If A is self-adjoint, then A − z is invertible for all z with Im z = 0, and
(A − z)−1 ≤

1
.
| Im z|

(2.6)

Since H is symmetric, it suffices to show that Ran(H + iλ) = H, for some
λ ∈ R, ±λ > 0, i.e. to show that the equation
(H + iλ)ψ = f

(2.7)

has a unique solution for every f ∈ H and some λ ∈ R, ±λ > 0. Write
2
H0 = − 2m Δ. We know H0 is self-adjoint, and so H0 + iλ is one-to-one and
onto, and hence invertible. Applying (H0 + iλ)−1 to (2.7), we find
ψ + K(λ)ψ = g,
where K(λ) = (H0 +iλ)−1 V and g = (H0 +iλ)−1 f . By (2.6), K(λ) ≤ λ1 V .
Thus, for |λ| > V , K(λ) < 1 and therefore 1 + K(λ) is invertible,

according to the first statement above. Similar statements hold also for
K(λ)T := V (H0 + iλ)−1 . Therefore
ψ = (1 + K(λ))−1 g
Moreover, it is easy to see that
(H0 + iλ)(1 + K(λ)) = (1 + K(λ)T )(H0 + iλ)
and therefore ψ = (H0 +iλ)−1 (1+K(λ)T ) (show this). So ψ ∈ D(H0 ) = D(H).
Hence Ran(H + iλ) = H and H is self-adjoint, by the second property above.
α
Unbounded potentials. The Coulomb potential V (x) = |x|
is not bounded.
We can extend the proof of Theorem 2.7 to show that Schră
odinger operators
with real potentials with Coulomb-type singularities are still self-adjoint. More
precisely, we consider a general class of potentials V satisfying for all ψ ∈
D(H0 )
(2.8)
V ψ ≤ a H0 ψ + b ψ

(H0 -bounded potentials) for some a and b with a < 1.
α
satisfies (2.8) with a > 0 arbitrary and
Problem 2.8 Show that V (x) = |x|
b depending on a. Hint : Write V (x) = V1 (x) + V2 (x) where

V1 (x) =

V (x)
0

|x| ≤ 1

,
|x| > 1

V2 (x) =

0
V (x)

|x| ≤ 1
|x| > 1.

Use that V1 ψ ≤ sup |ψ| V1 , that by the Fourier transform sup |ψ| ≤
( (|k|2 + c)−2 dk)−1/2 ( Δψ + c ψ ), and the fact that (|k|2 + c)−2 dk → 0
as c → ∞.

www.pdfgrip.com