www.TheSolutionManual.com HILARY D BREWSTER www.TheSolutionManual.com MATHEMATICAL PHYSICS www.TheSolutionManual.com "This page is Intentionally Left Blank" Hilary D Brewster Oxford Book Company Jaipur, India www.TheSolutionManual.com MATHEMATICAL PHYSICS ISBN: 978-93-80179-02-5 First Edition 2009 267, IO-B-Scheme, Opp Narayan Niwas, Gopalpura By Pass Road, Jaipur-302018 Phone: 0141-2594705, Fax: 0141-2597527 e-mail: oxfordbook@sify.com website: www.oxfordbookcompany.com © Reserved Typeset by: Shivangi Computers 267, 10-B-Scheme, Opp Narayan Niwas, Gopalpura By Pass Road, Jaipur-3020 18 Printed at: Rajdhani Printers, Delhi All Rights are 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, scanning or otherwise, without the prior written permission of the copyright owner Responsibility for the facts stated, opinions expressed, conclusions reached and plagiarism, ifany in this volume is entirely that of the Author, according to whom the matter encompassed in this book has been originally created/edited and resemblance with any such publication may be incidental The Publisher bears no responsibility for them, whatsoever www.TheSolutionManual.com Oxford Book Company This book is intended to provide an account of those parts of pure mathematics that are most frequently needed in physics This book will serve several purposes: to provide an introduction for graduate students not previously acquainted with the material, to serve as a reference for mathematical physicists already working in the field, and to provide an introduction to various advanced topics which are difficult to understand in the literature Not all the techniques and application are treated in the same depth In general, we give a very thorough discussion of the mathematical techniques and applications in quantum mechanics, but provide only an introduction to the problems arising in quantum field theory, classical mechanics, and partial differential equations This book is for physics students interested in the mathematics they use and for mathematics students interested in seeing how some of the ideas of their discipline find realization in an applied setting The presentation tries to strike a balance between formalism and application, between abstract and concrete The interconnections among the various topics are clarified both by the use of vector spaces as a central unifying theme, recurring throughout the book, and by putting ideas into their historical context Enough of the essential formalism is included to make the presentation self-contained This book features t~ applications of essential concepts as well as the coverage of topics in the this field Hilary D Brewster www.TheSolutionManual.com Preface www.TheSolutionManual.com "This page is Intentionally Left Blank" Preface l Mathematical Basics iii Laplace and Saddle Point Method 45 Free Fall and Harmonic Oscillators 67 Linear Algebra 107 Complex Representations of Functions 144 Transform Techniques in Physics 191 Problems in Higher Dimensions 243 Special Functions 268 Index 288 www.TheSolutionManual.com Contents www.TheSolutionManual.com "This page is Intentionally Left Blank" 276 Special Functions where ~ is the angle between'l and '2' Typically, one of the position vectors is larger than the other In this case, we have 'I « '2' So, one can write O Note that the Gamma function is undefined at zero and the negative integers Another useful formula is It is simply found as (1) = bF-.! r 2" Letting t = z2, we have t e-( dt www.TheSolutionManual.com r(x) ~ 283 Special Functions Due to the symmetry of the integrand, we obtain the classic integral r(~) = Ce-z\fz, which we had integrated when we computed the Fourier transform of a Gaussian Recall that z2 dz = f;c Therefore, we have confirmed that r( ~) = f;c We have seen that the factorial function can be written in terms of Gamma functions One can also relate the odd double factorials in terms of the Gamma function First, we note that (2n + I)! (2n)!! = 2nn!, (2n + I)!! = 2n , n In particular, one can prove r(n +.!.)2 = (211-1)!! In 2n Formally, this givesll (~} =r(%) ='If Another useful relation is 1t ['(x)r(l - x) =-.- SlOm This result can be proven using complex variable methods BESSEL FUNCTIONS Another important differential equation that arises in many physics applications is x 2y" + xy' + (x - p2)y = o This equation is readily put into self-adjoint form as (xy')' +( x 0,1 n=-oo *° Integral Representation p n=O n=1 n=2 n=3 n=4 n=5 I 2.405 3.832 5.135 6.379 7.586 8.780 5.520 7.016 8.147 9.760 11.064 12.339 8.654 10.173 11.620 13.017 14.373 15.700 11.792 13.323 14.796 16.224 17.616 18.982 14.931 16.470 17.960 19.410 20.827 22.220 18.071 19.616 21.I17 22.583 24.018 25.431 21.2 I 22.760 24.270 25.749 27.200 28.628 24.353 25.903 27.421 28.909 30.371 31.813 27.494 29.047 30.571 32.050 33.512 34.983 In(x) =; r cos(xsin8-n8)d8, x> 0, n E Z Fourier-Bessel Series Since the Bessel functions are an orthogonal set of eigenfunctions of a Sturm-Liouville problem, we can expand square integrable functions in this basis In fact, the eigenvalue problem is given in the form x 2y" + xy' + (Ax - p2)y O The solutions are then of the form JpC'Ex), as can be shown by making the substitution t = Ax in the differential equation Furthermore, if < x < a, and one solves the differential equation with boundary conditions that y(x) is bounded at x = and yea) = 0, then one can show that = 00 f(x) = LcnJp(Jpn x ) n=l where the Fourier-Bessel coefficients s are found using the ortho-gonality relation as www.TheSolutionManual.com Table: The Zeros of Bessel Functions Special Functions 287 Example: Expandf(x) = for < x < in a Fourier-Bessel series of the form 00 f(x) = LcnJo(Jpn x ) n=l We need only compute the Fourier-Bessel coefficients s in equatio~: c = 2 £xJo(JonX)dX n [J1(JOn)] rhn xJO(JOnx)dx = T.b yJo(y)dy JOn £ jon = _1_ r ~[yJl(Y)]dy ·2 b dy JOn =-[yJ (y)]JOn JOn =-.1- J1 (Jon) JOn As a result, we have found that the desired Fourier-Bessel expansion is 2~ JO(JOn x ) 1= L.J J( )' n=l JOn JOn O