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HILARY. D. BREWSTER MATHEMATICAL PHYSICS MATHEMATICAL PHYSICS Hilary. D. Brewster Oxford Book Company Jaipur, India ISBN: 978-93-80179-02-5 First Edition 2009 Oxford Book Company 267, IO-B-Scheme, Opp. Narayan Niwas, Gopalpura By Pass Road, Jaipur-302018 Phone: 0141-2594705, Fax: 0141-2597527 e-mail: [email protected] 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. Preface 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 Contents Preface iii l. Mathematical Basics 1 2. Laplace and Saddle Point Method 45 3. Free Fall and Harmonic Oscillators 67 4. Linear Algebra 107 5. Complex Representations of Functions 144 6. Transform Techniques in Physics 191 7. Problems in Higher Dimensions 243 8. Special Functions 268 Index 288 Chapter 1 Mathematical Basics Before we begin our study of mathematical physics, we should review some mathematical basics. It is assumed that you know Calculus and are comfortable with differentiation and integration. CALCULUS IS IMPORTANT There are two main topics in calculus: derivatives and integrals. You learned that derivatives are useful in providing rates of change in either time or space. Integrals provide areas under curves, but also are useful in providing other types of sums over continuous bodies, such as lengths, areas, volumes, moments of inertia, or flux integrals. In physics, one can look at graphs of position versus time and the slope (derivative) of such a function gives the velocity. Then plotting velocity versus time you can either look at the derivative to obtain acceleration, or you could look at the area under the curve and get the displacement: to x = vdt. Of course, you need to know how to differentiate and integrate given functions. Even before getting into differentiation and integration, you need to have a bag of functions useful in physics. Common functions are the polynomial and rational functions. Polynomial functions take the general form fix) = arfXn + all_1 xnn-1 + ... + a1x + ao' where an *-0.: This is the form of a polynomial of degree n. Rational functions consist of ratios of polynomials. Their graphs can exhibit asymptotes. Next are the exponential and logarithmic functions. The most common are the natural exponential and the natural logarithm. The natural exponential is given by fix) =~, where e:::: 2.718281828 .... The natural logarithm is the inverse to the exponential, denoted by In x. The properties of the expon~tial function follow from our basic properties for exponents. Namely, we have: