Mathematical methods for physicists

478 Pages·2020·4.471 MB·English

by  Babusci, Danilo, Dattoli, G., Licciardi, Silvia, Sabia, Elio

MATHEMATICAL
METHODS
FOR PHYSICISTS
11315_9789811201578_TP.indd 1 18/9/19 11:14 AM
June16,2015 14:37 BC:9335-Kernel-basedApproximationMethodsusingMATLAB FasshauerMcCourtBook pagevi
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
ATHEMATICAL
M
METHODS
FOR PHYSICISTS
Danilo Babusci
INFN, Italy
Giuseppe Dattoli
ENEA, Italy
Silvia Licciardi
ENEA, Italy
Elio Sabia
ENEA, Italy
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
11315_9789811201578_TP.indd 2 18/9/19 11:14 AM
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.
MATHEMATICAL METHODS FOR PHYSICISTS
Copyright © 2020 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 978-981-120-157-8
ISBN 978-981-120-268-1 (pbk)
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/11315#t=suppl
Desk Editor: Nur Syarfeena Binte Mohd Fauzi
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore
Syarfeena - 11315 - Mathematical Methods for Physicists.indd 1 18-09-19 11:44:13 AM
September18,2019 14:57 MathematicalMethodsforPhysicists 9.61inx6.69in b3642-main pagev
v
\If you intend to mount heavy mathematical artillery again during your com-
ing year in Europe, I would ask you not only not to come to Leiden, but if
possible not even to Holland, and just because I am really so fond of you and
want to keep it that way. But if, on the contrary, you want to spend at least
your (cid:12)rst few months patiently, comfortably, and joyfully in discussions that
keep coming back to the same few points, chatting about a few basic questions
with me and our young people - and without thinking much about publishing -
then I welcome you with open arms!! "
Eherenfest in a letter to Oppenheimer, Summer 1928
June16,2015 14:37 BC:9335-Kernel-basedApproximationMethodsusingMATLAB FasshauerMcCourtBook pagevi
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
September18,2019 14:57 MathematicalMethodsforPhysicists 9.61inx6.69in b3642-main pagevii
Contents
Abstract xiii
1 Matrices, Exponential Operators and Physical Applications 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Pauli Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Applications of 2(cid:2)2 Matrices . . . . . . . . . . . . . . . . . . 12
1.3.1 Classical Optics: Ray Beam Propagation and ABCD
Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Quantum Mechanics . . . . . . . . . . . . . . . . . . . 19
1.3.3 Particle Physics: Kaon Mixing . . . . . . . . . . . . . . 22
1.4 Cabibbo Angle and See-Saw Mechanism . . . . . . . . . . . . 29
1.5 Gell-Mann and Pauli Matrices . . . . . . . . . . . . . . . . . . 33
1.5.1 Spin Composition . . . . . . . . . . . . . . . . . . . . . 33
1.5.2 Gell-Mann Matrices . . . . . . . . . . . . . . . . . . . . 36
1.5.3 Flavors and Spin . . . . . . . . . . . . . . . . . . . . . 38
1.5.4 Color and QCD . . . . . . . . . . . . . . . . . . . . . . 41
1.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 43
1.6.1 Vector Di(cid:11)erential Equations and Matrices . . . . . . . 44
1.6.2 Matrices, Vector Equations and Rotations . . . . . . . 47
1.6.3 Cabibbo-Kobayashi-Maskawa Matrix . . . . . . . . . . 50
1.6.4 Frenet-Serret Equations . . . . . . . . . . . . . . . . . 51
1.6.5 Matrix, Rotations and Euler Angles . . . . . . . . . . . 52
1.6.6 4-Vectors and Lorentz Transformations . . . . . . . . . 53
1.6.7 Dirac Matrices . . . . . . . . . . . . . . . . . . . . . . 55
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
vii
September18,2019 14:57 MathematicalMethodsforPhysicists 9.61inx6.69in b3642-main pageviii
viii Mathematical Methods for Physicists
2 Ordinary and Partial Di(cid:11)erential Equations, Evolution
Operator Method and Applications 63
2.1 Ordinary Di(cid:11)erential Equations, Matrices and Exponential
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2 Partial Di(cid:11)erential Equations and Exponential
Operators, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3 Partial Di(cid:11)erential Equations and Exponential
Operators, II . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.4 Operator Ordering . . . . . . . . . . . . . . . . . . . . . . . . 73
2.5 Schro(cid:127)dinger Equation and Paraxial Wave Equation
of Classical Optics . . . . . . . . . . . . . . . . . . . . . . . . 79
2.6 Examples of Fokker-Planck, Schr(cid:127)odinger and Liouville
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 88
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3 Hermite Polynomials and Applications 101
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.2 Hermite Polynomials Generating Function . . . . . . . . . . . 104
3.2.1 Introducing the Generating Function . . . . . . . . . . 104
3.2.2 Generating Function Applications . . . . . . . . . . . . 106
3.3 Hermite Polynomials as an Orthogonal Basis . . . . . . . . . . 109
3.4 Hermite Polynomials in Quantum Mechanics: Creation and
Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . 113
3.5 Quantum Mechanics Applications . . . . . . . . . . . . . . . . 117
3.6 Coherent or Quasi-Classical States of Harmonic Oscillators . . 121
3.7 Jaynes-Cummings Model . . . . . . . . . . . . . . . . . . . . . 127
3.8 Classical Optics and Hermite Polynomials . . . . . . . . . . . 129
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4 Laguerre Polynomials, Integral Operators and
Applications 135
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2 Laguerre Polynomials Generating Function . . . . . . . . . . . 140
4.3 Orthogonality Properties of Laguerre Polynomials . . . . . . . 141
4.4 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.5 Associated Laguerre Polynomials . . . . . . . . . . . . . . . . 148
4.6 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . 151
September18,2019 14:57 MathematicalMethodsforPhysicists 9.61inx6.69in b3642-main pageix
Contents ix
4.7 Miscellaneous Applications and Comments . . . . . . . . . . . 153
4.8 App(cid:19)el Polynomials and Final Comments . . . . . . . . . . . . 158
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5 Exercises and Complements I 167
5.1 Pauli and Jones Matrices and Mueller Calculus . . . . . . . . 167
5.2 Magnetic Lenses and Matrix Description . . . . . . . . . . . . 174
5.3 Miscellanea on the Matrix Formalism and Solution
of Evolution Problems . . . . . . . . . . . . . . . . . . . . . . 182
5.3.1 Matrices and Quaternions . . . . . . . . . . . . . . . . 182
5.3.2 Matrix Solution of Evolution Problems . . . . . . . . . 183
5.4 Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . 186
5.5 Hyperbolic Trigonometry and Special Relativity . . . . . . . . 189
5.6 A Touch on Elliptic Functions . . . . . . . . . . . . . . . . . . 194
5.7 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . 207
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6 Exercises and Complements II 217
6.1 Ordinary Di(cid:11)erential Equations and Matrices . . . . . . . . . 217
6.2 Crofton-Glaisher Identities and Heat Type Equations . . . . . 230
6.3 Gamma Function and De(cid:12)nite Integrals . . . . . . . . . . . . . 235
6.4 Complex Variable Method and Evaluation of Integrals . . . . . 244
6.5 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 250
6.6 Fourier Transform and the Solution of Di(cid:11)erential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
6.7 Fourier-Type Transforms . . . . . . . . . . . . . . . . . . . . . 262
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
7 Exercises and Complements III 273
7.1 Second Solution of Hermite Equation . . . . . . . . . . . . . . 273
7.2 Higher Orders Hermite Polynomials . . . . . . . . . . . . . . . 275
7.3 Multi-Index Hermite Polynomials . . . . . . . . . . . . . . . . 280
7.4 Creation-Annihilation Operators Algebra and Physical
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
7.5 Eisenstein Integers . . . . . . . . . . . . . . . . . . . . . . . . 290
7.6 Harmonic Oscillator Hamiltonian Formal Aspects and Further
Miscellaneous Considerations . . . . . . . . . . . . . . . . . . 295
7.7 Time-Dependent Hamiltonians . . . . . . . . . . . . . . . . . . 298