Lectures on quantum mechanics for mathematics students

Lectures on Quantum Mechanics for Mathematics Students www.pdfgrip.com www.pdfgrip.com STUDENT MATHEMATICAL LIBRARY Volume 47 Lectures on Quantum Mechanics for Mathematics Students L D Faddeev A Yakubovskii Translated by Harold McFaden AMS American Mathematical Society www.pdfgrip.com Editorial Board Gerald B Folland Robin Forman Brad G Osgood (Chair) Michael Starbird The cover graphic was generated by Matt Strassler with help from Peter Skands Processed through CMS by Albert De Roeck, Christophe Saout and Joanna Weng Visualized by Ianna Osborne Copyright CERN 2000 Mathematics Subject Classification Primary 81-01, 8lQxx For additional information and updates on this book, visit www.ams.org/bookpages/stml-47 Library of Congress Cataloging-in-Publication Data Faddeev, L D [Lektsii po kvantovoi mekhanike dlia studentov-matematikov English] Lectures on quantum mechanics for mathematical students / L D Faddeev, O A Yakubovskii [English ed.] p cm - (Student mathematical library ; v 47) ISBN 978-0-8218-4699-5 (alk paper) Quantum theory I Iakubovskii, Oleg Aleksandrovich II Title QC174.125.F3213 2009 530.12-dc22 2008052385 Copying and reprinting Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 029042294, USA Requests can also be made by e-mail to reprint -permissionaams.org © 2009 by the American Mathematical Society All rights reserved The American Mathematical Society retains all rights except those granted to the United States Government Printed in the United States of America The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability Visit the AMS home page at http:1/wv.ams.org/ 10987654321 14131211 1009 www.pdfgrip.com Contents Preface ix Preface to the English Edition P The algebra of observables in classical mechanics xi §2 States §3 Liouville's theorem, and two pictures of motion in classical mechanics 13 §4 Physical bases of quantum mechanics 15 §5 A finite-dimensional model of quantum mechanics 27 States in quantum mechanics Heisenberg uncertainty relations Physical meaning of the eigenvalues and eigenvectors of 32 §6 §7 §8 observables §9 §10 §11 §12 §13 36 39 Two pictures of motion in quantum mechanics The 44 Schrodinger equation Stationary states Quantum mechanics of real systems The Heisenberg commutation relations 49 Coordinate and momentum representations 54 "Eigenfunctions" of the operators Q and P 60 The energy, the angular momentum, and other examples of observables 63 v www.pdfgrip.com Contents vi §14 §15 § 16 §17 § 18 §19 §20 §21 §22 §23 §24 §25 §26 §27 §28 §29 §30 §31 §32 §33 §34 §35 §36 The interconnection between quantum and classical mechanics Passage to the limit from quantum mechanics to classical mechanics 69 One-dimensional problems of quantum mechanics A free one-dimensional particle 77 The harmonic oscillator 83 The problem of the oscillator in the coordinate representation 87 Representation of the states of a one-dimensional particle in the sequence space 12 90 Representation of the states for a one-dimensional particle in the space D of entire analytic functions 94 The general case of one-dimensional motion 95 Three-dimensional problems in quantum mechanics A 103 three-dimensional free particle 104 A three-dimensional particle in a potential field 106 Angular momentum 108 The rotation group 111 Representations of the rotation group 114 Spherically symmetric operators Representation of rotations by x unitary matrices 117 Representation of the rotation group on a space of entire 120 analytic functions of two complex variables 123 Uniqueness of the representations Dj Representations of the rotation group on the space 127 L2(S2) Spherical functions 130 The radial Schrodinger equation 136 The hydrogen atom The alkali metal atoms 147 Perturbation theory 154 The variational principle Scattering theory Physical formulation of the problem 157 Scattering of a one-dimensional particle by a potential 159 barrier www.pdfgrip.com Contents vii Physical meaning of the solutions ik, and 02 Scattering by a rectangular barrier 164 Scattering by a potential center Motion of wave packets in a central force field The integral equation of scattering theory Derivation of a formula for the cross-section Abstract scattering theory Properties of commuting operators Representation of the state space with respect to a complete set of observables 169 §46 Spin 203 §47 Spin of a system of two electrons 208 §48 Systems of many particles The identity principle 212 Symmetry of the coordinate wave functions of a system of two electrons The helium atom 215 Multi-electron atoms One-electron approximation 217 223 The self-consistent field equations Mendeleev's periodic system of the elements 226 §37 §38 §39 §40 §41 §42 §43 §44 §45 §49 §50 §51 §52 Appendix: Lagrangian Formulation of Classical Mechanics www.pdfgrip.com 167 175 181 183 188 197 201 231 www.pdfgrip.com 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 mechanics was developed by the first author, who taught the course from 1968 to 1973 Subsequently the course was taught by the second author 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 common in the physics literature We take into account that the students not study general physics In a course intended for mathematicians, we have naturally aimed for a more rigorous presentation than usual of the mathematical questions in quantum mechanics, but not for full mathematical 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 Mechanics The present lectures differ essentially from Mackey's book both in the method of presentation of the bases of quantum mechanics and in the selection of material Moreover, these lectures assume somewhat less in the way of mathematical preparation of the students Nevertheless, we have borrowed much both from Mackey's ix www.pdfgrip.com § 50 Multi-electron atoms One-electron approximation 221 Let us consider the question of classifying the energy levels of a multi-electron atom The exact Schrodinger operator for the atom can be written in the form H=H'+WC+ Ws, where n WC = n i