Faddeev L.D., Yakubovskii O.A. Lectures on Quantum Mechanics for Mathematics Students
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Lectures on Quantum
Mechanics for
Mathematics Students
STUDENT MATHEMATICAL LIBRARY
Volume 47
Lectures on Quantum
Mechanics for
Mathematics Students
L. D. Faddeev
O. A. Yakubovskii
Translated by
Harold McFaden
AMS
American Mathematical Society
Editorial Board
Gerald B. Folland
Brad G Osgood (Chair)
Robin Forman :Michael Starbird
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2000 Mathematics Subject Classification. Primary 81-01, 8lQxx.
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Library of Congress Cataloging-in-Publication Data
Faddeev, L D
[T,ektsii po kvantovoi mekhanike dlia studentov-tnatematikov English]
Lectures on quantum mechanics for mathematical students / 1. D Facldeev,
0 A Yakubovskii
[English ed ]
p cin (Student mathematical library ; v 47)
ISBN 978-0-8218-4699-5 (alk paper)
1 Quantum theory
T Iakubovskii, Oleg Aleksandrovich TI Title
QC 174 125 F3213 2009
530 12 -dc22 2008052385
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Contents
Preface
Preface to the English Edition
ix
xi
§1. The algebra of observables in classical mechanics 1
§2.
States 6
§3-
Liouville's theorem, and two pictures of motion in
classical mechanics 13
§4.
Physical bases of quantum mechanics 15
§5.
A finite-diniensional model of quantum mechanics
27
§6.
States in quantum mechanics
32
§7.
Heisenberg uncertainty relations
36
§8
Physical meaning of the eigenvalues and eigenvectors of
observables 39
§9.
Two pictures of motion in quantum mechanics. The
Schrodinger equation Stationary states
44
§10.
Quantum mechanics of real systems. The Heisenberg
commutation relations
49
§11.
Coordinate and momentum representations 54
§12.
"Eigenfunctions" of the operators Q and P 60
§13.
The energy. the angular momentum, and other examples
of observables
63
v
Vi
Contents
§ 14 The interconnection between quantum and classical
mechanics.
Passage to the limit from quantum
rriechanics to classical mechanics 69
15. One-dimensional problems of quantum mechanics. A
free one-dimensional particle 77
D6
The harmonic oscillator 83
§17
The problem of the oscillator in the coordinate
representation 87
18.
Representation of the states of a one-dimensional
particle in the sequence space 12 90
§19.
Representation of the states for a one-dimensional
particle in the space V of entire analytic functions 94
§20.
The general case of one-dimensional motion
95
§21.
Three-dimensional problems in quantum mechanics. A
three-dimensional free particle
103
§22 A three-dimensional particle in a potential field
104
§23 Angular momentum
106
§24.
The rotation group
108
§25.
Representations of the rotation group
111
§26. Spherically symmetric operators
114
§27. Representation of rotations by 2 x 2 unitary matrices
117
§28. ]Representation of the rotation group on a space of entire
analytic functions of two complex variables
120
§29. Uniqueness of the representations D
123
§30.
Representations of the rotation group on the space
L 2 (S2). Spherical functions
127
§31
The radial Schrodinger equation
130
§32.
The hydrogen atom. The alkali metal atoms
136
§33.
Perturbation theory
147
§34.
The variational principle
154
§35.
Scattering theory. Physical formulation of the problem 157
§36
Scattering of a one-dimensional particle by a potential
barrier
1 59
Contents
vii
§37.
Physical meaning of the solutions V), and i'2
164
§38 Scattering by a rectangular barrier 167
§39.
Scattering by a potential center 169
§40. Motion of wave packets in a central force field 175
§41. The integral equation of scattering theory 181
§42. Derivation of a formula for the cross-section 183
§43. Abstract scattering theory 188
§44. Properties of commuting operators 197
§45. Representation of the state space with respect to a
complete set of observahles
201
§46. Spin 203
§47. Spin of a system of two electrons 208
§48. Systems of many particles. The identity principle 212
§49.
Symmetry of the coordinate wave functions of a system
of two electrons. The helium atom
215
§50.
Multi-electron atoms. One-electron approximation
217
§51.
The self-consistent field equations
223
§52.
Mendeleev's periodic system of the elements 226
Appendix: Lagrangian Formulation of Classical Mechanics
231
Preface
This textbook is a detailed survey of a course of lectures given in
the Mathematics-Mechanics Department of Leningrad University for
mathematics students. The program of the course in quantum me-
chanics was developed by the first author. who taught the course from
1968 to 1973. Subsequently the course was taught by the second au-
thor. It has certainly changed somewhat over these years, but its goal
remains the same: to give an exposition of quantum mechanics from
a point of view closer to that of a mathematics student than is com-
mon in the physics literature. We take into account that the students
do not study general physics. In a course intended for mathemati-
cians. we have naturally aimed for a more rigorous presentation than
usual of the mathematical questions in quantum mechanics, but not
for full matherriatical rigor, since a precise exposition of a number of
questions would require a course of substantially greater scope.
In the literature available in Russian, there is only one book
pursuing the same goal, and that is the American mathematician
G. W. Mackey's book, Mathematical Foundations of Quantum Me-
chanics. The present lectures differ essentially from Mackey's book
both in the method of presentation of the bases of quantum mechan-
ics and in the selection of material. Moreover, these lectures assume
somewhat less in the way of mathematical preparation of the stu-
dents.
Nevertheless, we have borrowed much both from Mackey's
ix