Adiabatic Perturbation Theory in Quantum Dynamics
Separation of scales plays a fundamental role in the understanding of the dynamical behaviour of complex systems in physics and other natural sciences. A prominent example is the Born-Oppenheimer approximation in molecular dynamics. This book focuses on a
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Stefan Teufel
Adiabatic Perturbation Theory in Quantum Dynamics
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Author Stefan Teufel Zentrum Mathematik Technische Universit¨at M¨unchen Boltzmannstr. 3 85747 Garching bei M¨unchen, Germany e-mail: [email protected]
Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2000): 81-02, 81Q15, 47G30, 35Q40 ISSN 0075-8434 ISBN 3-540-40723-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm 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-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH springer.de c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10951750
41/3142/du-543210 - Printed on acid-free paper
Table of Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The time-adiabatic theorem of quantum mechanics . . . . . . . . . 1.2 Space-adiabatic decoupling: examples from physics . . . . . . . . . . 1.2.1 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 The Dirac equation with slowly varying potentials . . . . 1.3 Outline of contents and some left out topics . . . . . . . . . . . . . . . .
1 6 15 15 21 27
2
First order adiabatic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The classical time-adiabatic result . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Perturbations of fibered Hamiltonians . . . . . . . . . . . . . . . . . . . . . 2.3 Time-dependent Born-Oppenheimer theory: Part I . . . . . . . . . . 2.3.1 A global result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Local results and effective dynamics . . . . . . . . . . . . . . . . 2.3.3 The semiclassical limit: first remarks . . . . . . . . . . . . . . . . 2.3.4 Born-Oppenheimer approximation in a magnetic field and Berry’s connection . . . . . . . . . . . . .
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