Bose, spin and Fermi systems : problems and solutions

381 Pages·2015·21.257 MB·English

by  Hardy, Yorick, Steeb, Willi-Hans

_
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