HILARY. D. BREWSTER MATHEMATICAL PHYSICS "This page is Intentionally Left Blank" 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: 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. 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 "This page is Intentionally Left Blank" 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 "This page is Intentionally Left Blank" 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: x = to 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 + a ll _ 1 xn n - 1 + + a 1 x + 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: [...]... :: d; / ,=! 1 1 The differentiation of vectors yield is given by the following expressions: - dU _{dU! - - - 3 - - - - ,dU 2 ,dU- } dt dt dt dt T dU dt ' I ::::::> i =1,2,3 i.e each component of the vector is included in the differentiation As the considered velocity vector depends on space xi and the time t, the following differentiation law holds: dU· = + (dX ) au1 _ _ au· 1 1 I dt at ax i dt... the product rule for differentiation The two formulae are related by using the relations uj(x) ~ du = I(x) dx, u g(x) ~ dv = g'(x) dx This also gives a method for applying the Integration by Parts Formula Example: Consider the integral Jx sin 2x dx We choose u = x and dv = sin 2x dx This gives the correct left side of the formula We next determine v and du: du du= -dx=dx dx ' v = fdv = fsin2x dx = -%COS2X... of zero order (scalars), tensors of first order (vectors) and tensors of second order are employed in fluid mechanics to describe the fluid flows and usually attributed to Euler (1707-1783) In this description all quantities considered in the presentations of fluid-mechanics are dealt as functions of space and time Mathematical operations like addition, subtraction, division, multiplication, differentiation,... products often the so called Einstein's summation convention is applied By this one understands the summation over the same indices in a product When forming a product from tensors, one distinguishes the outer product and the inner product The outer product is again a tensor, where each element ofthe first tensor multiplied with each element of the second tensor results in an element of the 22 Mathematical. .. tensor of second order by the unit tensor of second order, i.e the 'Kronecker Delta', yields the initial tensor of second order {8ij}.{aij}={~ ~ ~}.{::: o 0 I a31 ::: :::}={aij} a32 a33 Further products can be formulated, as for example cross products between vectors and tensors of second order {a/} {b jk } =Eikl ·ai ·b Jk but these are not of special importance for the laws in fluid mechanics Mathematical. .. are presented as field variables and are thus functions of space and time It is assumed that in each space the thermodynamics connections between the state quantities hold, a~ for example the state equations that can be formulated for thermodynamically ideal fluids as follows p = const (state equation of the thermodynamically ideal liquids)P /p=RT (state equation of the thermodynamically ideal gases)... quantities with different units must not be added or subtracted vectorially For the addition and subtraction of vectorial constants (having the same units) the following rules of addition hold: Mathematical Basics 18 a + 0 = {a/ } + {O} =a (neutral elements 0) a+{-a} = {a;} + {-a;} =0 (a element inverse to -a) a +b = b + a, d. h {a;} + {bd = {b;} + {ad = {(aj +bj)} (commutative law) a+(b +c)=(a+b)+c ,d. h.{a i... product and the vector product leads to the scalar-triple-product (STP) formed of three vectors [aJ,cJ=a.(b- xc) The properties of this product from three vectors can be seen from the sketch shown below The scalar-triple-product (STP) of the vectors a,f),c leads to the six times the volume of parallelopiped (ppd), V ppd defined by the vectors a,b andc Fig Graphical Representation of Scalar-triple-product... undefined 1 1t 3 2 4 2 2 2 You also learned that they are represented as the ratios of the opposite to hypotenuse, adjacent to hypotenuse, etc Hopefully, you have this down by now You should also know the exact values for the special 1t 1t 1t 1t angles e = 0, 6"' 3' 4' 2" ' and their corresponding angles in the second, third and fourth quadrants This becomes internalized after much use, but we provide... the mass forces and mass acceleration acting locally on the fluid Thus the velocity OJ =U J (x ,I) , the rotation wJ I =w- j (x i ,I) , the force 24 Mathematical Basics Kj =Kj(xl't) and the acceleration gj(x"t) can be stated as field quantities and be employed as such quantities in the following considerations Analogously tensors of second and higher order can also be introduced as field variables For . 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. this rule tells us that if we have a composition of functions, such as the elementary functions above, then we can compute the derivative of the composite function. Namely, if hex). might be happier using a computer with a computer algebra systems, such as Maple, you should know a few basic integrals and know how to use tables for some of the more complicated ones. In fact,