_ W illi-H a n s S te e b H a r d y Y o rick Bose, Spin and Fermi Systems P R O B L E M S AND SO LU T IO N S This page intentionally left blank P R O B L E M S AND S O L U T I O N S W illi-Hans Steeb Yorick H ardy University of Johannesburg and University of South Africa, South Africa World Scientific NEW JERSEY • LONDON • SI NGAPORE • BEI JI NG • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. BOSE, SPIN AND FERMI SYSTEMS Problems and Solutions Copyright © 2015 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 o f 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 978-981-4630-10-8 ISBN 978-981-4667-34-0 (pbk) Printed in Singapore P r e f a c e The purpose of this book is to supply a collection of problems and solutions for Bose, spin and Fermi systems as well as coupled systems. So it covers essential parts of quantum theory and quantum field theory. For most of the problems the detailed solutions are provided which will prove to be valuable to graduate students as well as to research workers in these fields. Each chapter contains supplementary problems often with the solution provided. All the important concepts are provided either in the introduction or the problem and all relevant definitions are given. The topics range in difficulty from elementary to advanced. Almost all problems are solved in detail and most of the problems are self-contained. Students can learn important principles and strategies required for problem solving. Teachers will also find this text useful as a supplement, since important concepts and techniques are developed in the problems. The book can also be used as a text or a supplement for quantum theory, Hilbert space theory and linear and multilinear algebra or matrix theory. Computer algebra programs in Symbol- icC-b-b and Maxima are also included. For Bose system number states, coherent states and squeezed states are covered. Applications to nonlinear dynamical systems and linear optics are given. The spin chapter concentrates mostly on spin-| and spin-1 systems, but also higher order spins are included. The eigenvalue problem plays a central role. Exercises utilizing the spectral theorem and Cayley-Hamilton theorem are provided. For Fermi systems a special section on the Hubbard Hamilton operator is added. Chapter 4 is devoted to Lie algebras and their representation by Bose, Spin and Fermi operators. Superalgebras are also consid­ ered. Chapters 5 and 6 cover coupled Bose-Spin and coupled Bose-Fermi systems, respectively. The material was tested in our lectures given around the world. Any useful suggestions and comments are welcome. The International School for Scientific Computing (ISSC) provides certificate courses for this subject. Please contact the first author if you want to do this course. More exercises can be found on the web page given below. e-mail addresses of the authors: steebwilliOgmail. com yorickhardyOgmail. com Home page of the first author: http://issc.uj.ac.za v This page intentionally left blank C o n t e n t s 1 B ose System s 1 1.1 Commutators and Number S ta te s ........................................................................... 1 1.2 Coherent S ta te s ............................................................................................................. 25 1.3 Squeezed S ta te s ............................................................................................................. 47 1.3.1 One-Mode Squeezed S t a t e s ............................................................................ 47 1.3.2 Two-Mode Squeezed S t a t e s ............................................................................ 53 1.4 Coherent Squeezed S t a t e s ........................................................................................... 59 1.5 Hamilton O p e r a to r s .................................................................................................... 62 1.6 Linear O p tic s ................................................................................................................ 74 1.7 Classical Dynamical S y s te m s ..................................................................................... 78 1.8 Supplementary Problem s.............................................................................................. 82 2 Spin System s 94 2.1 Spin Matrices, Commutators and A nticom m utators.................................................94 2.2 Spin Matrices and Functions ................................................................................... 135 2.3 Spin Hamilton Operators ......................................................................................... 153 2.4 Supplementary Problem s............................................................................................ 173 3 Fermi System s 194 3.1 States, Anticommutators, C o m m u ta to rs................................................................. 195 3.2 Fermi Operators and F u n c tio n s................................................................................ 216 3.3 Hamilton O p e r a to r s .................................................................................................. 228 3.4 Hubbard M o d e l........................................................................................................... 247 3.5 Supplementary Problem s............................................................................................ 259 4 Lie Algebras 270 4.1 Lie Algebras and Bose O p e ra to rs ............................................................................. 270 4.2 Lie Algebras and Spin O p e r a to r s ............................................................................. 287 4.3 Lie Algebras and Fermi O p e r a to r s .......................................................................... 297 4.4 Lie S u p e ralg e b ra ........................................................................................................ 304 4.5 Supplementary Problem s............................................................................................ 309 5 B ose-Spin System s 318 5.1 Solved P r o b le m s ........................................................................................................ 318 5.2 Supplementary Problem s............................................................................................ 333 6 Bose-Ferm i System s 337 6.1 Solved P r o b le m s ........................................................................................................ 337 6.2 Supplementary Problem s............................................................................................ 349 V•l l• viii Contents Bibliography 352 Index 365 N o t a t i o n 0 empty set A c B subset A of set B A n B the intersection of the sets A and B A l l B the union of the sets A and B maps f->9 composition of two mappings ( / o g)(x) = f(g(x)) f ° 9 N natural numbers No natural numbers including 0 Z integers Q rational numbers R real numbers R+ nonnegative real numbers C complex numbers Mn n-dimensional Euclidian space Cn n-dimensional complex linear space lm := z complex number z, z* complex conjugate of z $lz real part of the complex number z $sz imaginary part of the complex number z v Cn element v of Cn (column vector) G y* transpose and complex conjugate of v v*v scalar product 0 zero vector (column vector) t time variable u frequency X space variable VT = (ui, Vn) vector of independent variables, T means transpose norm x • y = x Ty scalar product (inner product) in vector space Rn x x y vector product in vector space R3 det determinant of a square matrix tr trace of a square matrix n x n zero matrix On n x n unit matrix (identity matrix) In A T transpose of matrix A A * transpose and complex conjugate of matrix A identity operator I commutator IX