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Classical and Quantum Mechanics of Noncentral Potentials A Survey of Two-Dimensional Systems

R.S. Kaushal

Springer Science+Business Media, LLC

Dr. R.S. Kaushal Department of Physics and Astrophysics University of Delh~ Delhi-110 007. India Copyright C> 1998 Springer Science+Business Media New Y ru:k Originally published by Springer-Verlag Berlin Heidelberg New York in 1998 All rights reserved. No part of this publication may be reproduced, Stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical. photocopying, recording or otherwise, without the prior permission of the publisher. Exclusive distribution in Europe and North America (including Canada, Mexico and South America) by Springer-V erlag Berlin Heidelberg GmbH. All export rights for this book vest exclusively with Narosa Publishing House. Unauthorised export is a violation of Copyright Law and is subject to legal action.

ISBN 978-3-662-11327-1 ISBN 978-3-662-11325-7 (eBook) DOI 10.1007/978-3-662-11325-7

To My Revered Father Late Sri Kshem Karan Who Left a Number of Good Books at Home at the Time of his Departure for his Heavenly abode.

Na sato vidyate bha vo nabhavo vidyate satah, Ubhayorapi drastoantastvanayostatva darsibhih. - Srimad Bhagvad-Gita (2/16) The unreal has no existence, and the real never ceases to be; the reality of both (real and unreal) has thus been perceived by the seers of Truth.

Preface In different branches of science and engineering there appear now several newly discovered phenomena whose theoretical Wlderstanding is d.efinitely going to require an account not only of anharmonic but also of noncentral nature of the underlying forces. This book is an attempt to bring the available knowledge concerning some of the important classical and quantum aspects of such noncentral forces (potentials) in two dimensions at one place. With a view to provide readymade results for immediate applications, time dependent and time independent systems are studied separately at the level of both classical and quantum mechanics. While some of the topics are discussed at a great 1~ (such as the construction of invariants in the classical domain and the solution of the Schrodinger eqUation for such potentials in the quantum domain) others (such as Berry's phase, classical and quantum chaos, group- and field-theoretic aspects, extension of the results derived for 20 to the case of the corresponding 3D systems etc.) are given only a passing touch in the· form of either a Section in the Chapter or an Appendix. I accept that by doing so I have not done justice to these topics which are growing rather exponentially these days but such limitations arose mainly by keeping the size of the book in mind and secondly, most of these topics are already discussed in other specialized books. This latter types of topics are, however, presented here only in an introductory style and that too for the sake of completeness. Again, as several reviews and books are now available on the classical mechanics of noncentral forces, par