Problems and Solutions on Quantum Mechanics

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1 Problems and Solutions on Quantum Mechanics

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3 Major American Universities Ph. D. Qualifying Questions and Solutions Problems and Solutions on Quantum Mechanics Compiled by: The Physics Coaching Class University of Science and Technology of China Edited by: Yung-Kuo Lim V S% World Scientific wl Singapore'NewJersey L Singapore New Jersey 'London* Hong Kong

4 Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore USA office: Suite IB, 1060 Main Street, River Edge, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Major American Universities Ph.D. Qualifying Questions and Solutions PROBLEMS AND SOLUTIONS ON QUANTUM MECHANICS Copyright 1998 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN ISBN (pbk) This book is printed on acid-free paper. Printed in Singapore by Uto-Print

PREFACE This series of physics problems and solutions, which consists of seven volumes - Mechanics, Electromagnetism, Optics, Atomic, Nuclear and Particle Physics, Thermodynamics and Statistical

5 PREFACE This series of physics problems and solutions, which consists of seven volumes - Mechanics, Electromagnetism, Optics, Atomic, Nuclear and Particle Physics, Thermodynamics and Statistical Physics, Quantum Mechanics, Solid State Physics and Relativity, contains a selection of 2550 problems from the graduate-school entrance and qualifying examination papers of seven U.S. universities - California University Berkeley Campus, Columbia University, Chicago University, Massachusetts Institute of Technology, New York State University Buffalo Campus, Princeton University, Wisconsin University - as well as the CUSPEA and C. C. Ting's papers for selection of Chinese students for further studies in U.S.A., and their solutions which represent the effort of more than 70 Chinese physicists plus some 20 more who checked the solutions. The series is remarkable for its comprehensive coverage. In each area the problems span a wide spectrum of topics, while many problems overlap several areas. The problems themselves are remarkable for their versatility in applying the physical laws and principles, their uptodate realistic situations, and their scanty demand on mathematical skills. Many of the problems involve order-of-magnitude calculations which one often requires in an experimental situation for estimating a quantity from a simple model. In short, the exercises blend together the objectives of enhancement of one's understanding of the physical principles and ability of practical application. The solutions as presented generally just provide a guidance to solving the problems, rather than step-by-step manipulation, and leave much to the students to work out for themselves, of whom much is demanded of the basic knowledge in physics. Thus the series would provide an invaluable complement to the textbooks. The present volume consists of 380 problems. It covers practically the whole of the usual undergraduate syllabus in quantum mechanics, just falling short of quantum field theory, but in substance and sophistication going much beyond. In editing, no attempt has been made to unify the physical terms, units and symbols. Rather, they are left to the setters' and solvers' own preference so as to reflect the realistic situation of the usage today. Great pains has been taken to trace the logical steps from the first principles to the final solution, frequently even to the extent of rewriting the entire solution.

6 VI Preface In addition, a subject index to problems has been included to facilitate the location of topics. These editorial efforts hopefully will enhance the value of the volume to the students and teachers alike. The editor is most grateful to Prof. C. H. Oh of the National University of Singapore for some most illuminating discussion on the topics. Yung-Kuo Lira Editor

7 INTRODUCTION Solving problems in school work is exercise of the mind and enhances understanding of the principles. In general examination questions usually parallel such problems. Thus working out problems forms an essential and important part of the study of physics. Major American University Ph.D. Qualifying Questions and Solutions is a series of seven volumes. The subjects of each volume and the respective referees (in parentheses) are as follows: 1. Mechanics (Qiang Yuan-qi, Gu En-pu, Cheng Jia-fu, Li Ze-hua, Yang De-tian) 2. Electromagnetism (Zhao Shu-ping, You Jun-han, Zhu Jun-jie) 3. Optics (Bai Gui-ru, Guo Guang-can) 4. Atomic, Nuclear and Particle Physics (Jin Huai-cheng, Yang Baozhong, Fan Yang-mei) 5. Thermodynamics and Statistical Physics (Zheng Jiu-ren) 6. Quantum Mechanics (Zhang Yong-de, Zhu Dong-pei, Fan Hong-yi) 7. Solid State Physics, Relativity and Miscellaneous Topics (Zhang Jialu, Zhou You-yuan, Zhang Shi-ling) This series covers almost all aspects of University Physics and contains 2550 problems, most of which are solved in detail. The problems have been carefully chosen from a collection of 3100 problems, of which some came from the China-U.S.A. Physics Examination and Application (CUSPEA) Program, some were selected from the Ph.D. Qualifying Examination on Experimental High Energy Physics sponsored by Chao Chong Ting. The rest came from the graduate preliminary or qualifying examination questions of seven world-renowned American universities: Columbia University, University of California at Berkeley, Massachusetts Institute of Technology, University of Wisconsin, University of Chicago, Princeton University and State University of New York, Buffalo. Generally speaking, examination problems in physics in American universities do not involve too much mathematics. Rather, they are to a large extent characterized by the following three aspects. Some problems involving various frontier subjects and overlapping domains of science are selected by professors directly from their own research work and thus have an "upto-date" flavor. Some problems involve broad fields and require a quick vii

8 vm Introduction mind to analyse, while the methods needed for solving the other problems are simple and practical but requires a full "touch of physics". Indeed, we venture to opine that the problems, as a whole, embody to some extent the characteristics of American science and culture, as well as the philosophy underlying American education. Therefore, we considered it worthwhile to collect and solve these problems and introduce them to students and teachers, even though the effort involved was formidable. As many as a hundred teachers and graduate students took part in this time-consuming task. A total of 380 problems make up this volume of eight parts: basic principles and one-dimensional motions (72), central potentials (27), spin and angular momentum (48), motion in electromagnetic field (16), perturbation theory (83), scattering theory and quantum transitions (61), many-particle systems (37), and miscellaneous topics (40). In scope and depth, most of the problems conform to undergraduate physics syllabi for quantum mechanics in most universities, while many are rather profound and sophisticated or broad-based. A remarkable fact is that the problems from American universities often combine fundamental principles and latest research activities. Thus the problems may help the reader not only to enhance understanding of basic principles, but also to cultivate the ability of solving practical problems in a realistic environment. This volume is the result of the collective effort of 19 physicists who worked out and checked the solutions, notably Zhang Yong-de, Zhu Dongpei, Fan Hong-yi, Ren Yong, Dai Tie-sheng, Ning Bo. The original translation was carried out by professors Zheng Jiu-ren and Qi Bo-yun.

9 CONTENTS Preface Introduct Parti Part II Part III Part IV Part V Part VI Part VII Part VIII Index to Problems ion Basic Principles and One-dimensional Motions ( ) Central Potentials ( ) Spin and Angular Momentum ( ) Motion in Electromagnetic Fields ( ) Perturbation Theory ( ) Scattering Theory and Quantum Transitions ( ) Many-Particle Systems ( ) Miscellaneous Topics ( ) V vii

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