Analytic Methods in Physics

Charlie Harper

Analytic Methods in Physics. Charlie HarperCopyright © 1999 WILEY-VCH Verlag Berlin GmbH, BerlinISBN: 3-527-40216-0

Charlie Harper

Analytic Methodsin Physics

)W1LEY-VCHBerlin • Weinheim • New York • Chichester • Brisbane • Singapore • Toronto

Author:

Prof. Dr. Charlie Harper, California State University, Hayward, USA

With 96 figs, and 19 tabs.

1st edition

Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Harper, Charlie : Analytic methods in physics : with 19 tables / Charlie Harper. -1. ed. - Berlin ; Weinheim ; New York ; Chichester ; Brisbane ; Singapore ; Toronto :WILEY-VCH, 1999

ISBN 3-527-40216-0

This book was carefully produced. Nevertheless, authors, editors, and publishers do not warrant theinformation contained therein to be free of errors. Readers are advised to keep in mind that state-ments, data, illustrations, procedural details, or other items may inadvertently be inaccurate.All rights reserved (including those of translation into other languages). No part of this book maybe reproduced in any form - by photoprinting, microfilm, or any other means - nor transmitted ortranslated into a machine language without written permission from the publishers. Registerednames, trademarks, etc. used in this book, even when not specifically marked as such, are not to beconsidered unprotected by law.

© WILEY-VCH Verlag Berlin GmbH, Berlin (Federal Republic of Germany), 1999

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WILEY-VCH Verlag Berlin GmbHBuhringstraBe 10D-13086 BerlinFederal Republic of Germany

Preface

This book is intended for undergraduate students who are majoring in physics (or otherphysical sciences), applied mathematics, or engineering. The goal of the book is to providethe essential mathematical physics background needed for the study of analytic mechan-ics and mechanical wave motion, heat and thermodynamics, electromagnet ism, modernphysics, and quantum mechanics. In addition, it is intended to provide the necessary back-ground for advanced work in these areas. Numerous examples and illustrations are giventhroughout each chapter, and problems are included at the end of each chapter. Certainproblems are used to introduce new material in a self-contained manner.Throughout the book, the level of presentation is fairly uniform (with the possible exceptionof certain sections of chapters 2 and 12), and the included material should be readilyaccessible to undergraduate students who have a working knowledge of general physics andof differential and integral calculus. The desire to maintain a uniform level of presentationwas used as a guide for the selection of topics to include (and exclude).The material on essentials of vector spaces, essential algebraic structures, and exteriordifferential forms is invaluable for the study of physics today; the inclusion and presentationof this material at the undergraduate level are among the distinguishing features of thisbook.Chapter 1 contains a standard treatment of vector analysis, fundamental physical quanti-ties with SI units and dimensional analysis, and vector quantities in orthogonal curvilinearcoordinates and coordinate transformations. Also, Maxwell's equations in both differen-tial and integral forms are included, as well as an introduction of the notion of a gaugetransformation.In Chapter 2, a standard treatment of matrix analysis is presented. Then, essentialsof vector spaces including notions of a function (mapping), a linear operator, eigenvaluesand eigenfunctions, and the matrix representation of a linear operator are covered. Also,a brief introduction of topological spaces and elementary definitions of Hausdorff space,Banach space, Hilbert space, a manifold and topology is presented. The developmenton essential algebraic structures contains an elementary introduction to groups, rings, andfields as needed in mathematical physics. The primer on group theory in physics is requiredbackground for further study of applications of groups in physics.Chapters 3-6 are standard treatments of functions of a complex variable, the calculus ofresidues, Fourier series, and Fourier transforms. These topics are introduced early so thatthese concepts will be available for use when needed in subsequent chapters, such as inChapters 7 and 8.Chapters 7 and 8 on ordinary and partial differentials contain a standard treatment ofthese subjects; in addition, algorithms for the numerical solutions of ordinary and partialdifferential equations are developed.

6 PREFACE

Chapter 9 on special functions begins with the Sturm-Liouville theory and orthogonalpolynomials. It is shown that the completeness and orthogonality relations lead to theexpansion of a function in terms of orthogonal polynomials or orthogonal functions; Fourierseries, Legendre series, and the Hermite series are developed as examples of such expansions.The development of special functions via the power series solutions of appropriate ordinarydifferential equations follows the Sturm-Liouville theory. Also, the connections of specialfunctions with the hypergeometric functions or the confluent hypergeometric functions arediscussed.A brief and elementary introduction of integral equations is given in Chapter 10; this chapterends with the famous Abel problem.The introduction and applications of the calculus of variations and elementary functionalanalysis are given in Chapter 11. Hamilton's variational principle, Lagrangian and Hamil-tonian mechanics are treated in some details. Also, the transition from classical mechanicsto quantum mechanics in the Heisenberg picture, Schrodinger picture, and in the Feynmanpath integral approach is discussed.Chapter 12 begins with a brief overview of differential geometry, differentiate manifold,and coordinate transformations in linear spaces. Then, a treatment of standard tensoranalysis using indices ends with the Einstein equation in general relativity. This is followedby a coordinate free treatment of tensors involving exterior differential forms.I am grateful to the many physics students who used many sections of the manuscript.

Charlie Harper

Contents

1 Vector Analysis 131.1 Introduction 13

1.1.1 Background 131.1.2 Properties and Notations 141.1.3 Geometric Addition of Vectors 14

1.2 The Cartesian Coordinate System 161.2.1 Orthonormal Basis Vectors: i, j, k 161.2.2 Rectangular Resolution of Vectors 161.2.3 Direction Cosines 181.2.4 Vector Algebra 19

1.3 Differentiation of Vector Functions 261.3.1 The Derivative of a Vector Function 261.3.2 Concepts of Gradient, Divergence, and Curl 27

1.4 Integration of Vector Functions 311.4.1 Line Integrals 321.4.2 The Divergence Theorem Due to Gauss 341.4.3 Green's Theorem 401.4.4 The Curl Theorem Due to Stokes 41

1.5 Orthogonal Curvilinear Coordinates 431.5.1 Introduction 431.5.2 The Gradient in Orthogonal Curvilinear Coordinates 461.5.3 Divergence and Curl in Orthogonal Curvilinear

Coordinates 461.5.4 The Laplacian in Orthogonal Curvilinear Coordinates 471.5.5 Plane Polar Coordinates (r,0) 471.5.6 Right Circular Cylindrical Coordinates (p, 0, z) 471.5.7 Spherical Polar Coordinates (r,0,0) 48

1.6 Problems 491.7 Appendix I: Systeme International (SI) Units 521.8 Appendix II: Properties of Determinants 53

1.8.1 Introduction 531.8.2 The Laplace Development by Minors 55

1.9 Summary of Some Properties of Determinants 56

8 CONTENTS

2 Modern Algebraic Methods in Physics 592.1 Introduction 592.2 Matrix Analysis 60

2.2.1 Matrix Operations 612.2.2 Properties of Arbitrary Matrices 632.2.3 Special Square Matrices 642.2.4 The Eigenvalue Problem 682.2.5 Rotations in Two and Three Dimensions 69

2.3 Essentials of Vector Spaces 712.3.1 Basic Definitions 712.3.2 Mapping and Linear Operators 722.3.3 Inner Product and Norm 742.3.4 The Legendre Transformation 752.3.5 Topological Spaces 762.3.6 Manifolds 79

2.4 Essential Algebraic Structures 802.4.1 Definition of a Group 802.4.2 Definitions of Rings and Fields 812.4.3 A Primer on Group Theory in Physics 82

2.5 Problems 89

3 Functions of a Complex Variable 953.1 Introduction 953.2 Complex Variables and Their Representations 953.3 The de Moivre Theorem 983.4 Analytic Functions of a Complex Variable 993.5 Contour Integrals 1023.6 The Taylor Series and Zeros of f(z) 106

3.6.1 The Taylor Series 1063.6.2 Zeros of f(z) 108

3.7 The Laurent Expansion 1083.8 Problems 1133.9 Appendix: Series 115

3.9.1 Introduction 1153.9.2 Simple Convergence Tests 1163.9.3 Some Important Series in Mathematical Physics 116

4 Calculus of Residues 1194.1 Isolated Singular Points 1194.2 Evaluation of Residues 121

4.2.1 m-th-Order Pole 1214.2.2 Simple Pole 121

4.3 The Cauchy Residue Theorem 1254.4 The Cauchy Principal Value 1264.5 Evaluation of Definite Integrals 127

4.5.1 Integrals of the Form /0^/(sin0,cos0)d0 1274.5.2 Integrals of the Form f!°00f(x)dx 128

CONTENTS 9

4.5.3 A Digression on Jordan's Lemma 1304.5.4 Integrals of the Form /^ f(x)eimxdx 131

4.6 Dispersion Relations 1324.7 Conformal Transformations 1344.8 Multi-valued Functions 1374.9 Problems 141

5 Fourier Series 1435.1 Introduction 1435.2 The Fourier Cosine and Sine Series 1445.3 Change of Interval 1445.4 Complex Form of the Fourier Series 1455.5 Generalized Fourier Series and the Dirac Delta Function 1495.6 Summation of the Fourier Series 1515.7 The Gibbs Phenomenon 1535.8 Summary of Some Properties of Fourier Series 1545.9 Problems 155

6 Fourier Transforms 1576.1 Introduction 1576.2 Cosine and Sine Transforms 1596.3 The Transforms of Derivatives 1626.4 The Convolution Theorem 1646.5 Parseval's Relation 1656.6 Problems 166

7 Ordinary Differential Equations 1677.1 Introduction 1677.2 First-Order Linear Differential Equations 168

7.2.1 Separable Differential Equations 1687.2.2 Exact Differential Equations 1697.2.3 Solution of the General Linear Differential Equation 170

7.3 The Bernoulli Differential Equation 1737.4 Second-Order Linear Differential Equations 174

7.4.1 Homogeneous Differential Equations with Constant Coefficients . . . 1757.4.2 Nonhomogeneous Differential Equations with Constant Coefficients . 1797.4.3 Homogeneous Differential Equations with Variable Coefficients . . . . 1827.4.4 Nonhomogeneous Differential Equations with Variable Coefficients . . 184

7.5 Some Numerical Methods 1867.5.1 The Improved Euler Method for First-Order Differential Equations . 1867.5.2 The Runge-Kutta Method for First-Order Differential Equations . . . 1887.5.3 Second-Order Differential Equations 189

7.6 Problems 189

10 CONTENTS

8 Partial Differential Equations 1958.1 Introduction 1958.2 The Method of Separation of Variables 197

8.2.1 The One-Dimensional Heat Conduction Equation 2008.2.2 The One-Dimensional Mechanical Wave Equation 2018.2.3 The Time-Independent Schrodinger Wave Equation 205

8.3 Green's Functions in Potential Theory 2068.4 Some Numerical Methods 208

8.4.1 Fundamental Relations in Finite Differences 2088.4.2 The Two-Dimensional Laplace Equation: Elliptic Equation 2088.4.3 The One-Dimensional Heat Conduction Equation: Parabolic Equation 2088.4.4 The One-Dimensional Wave Equation: Hyperbolic Equation 209

8.5 Problems 210

9 Special Functions 2159.1 Introduction 2159.2 The Sturm-Liouville Theory 216

9.2.1 Introduction 2169.2.2 Hermitian Operators and Their Eigenvalues 2189.2.3 Orthogonality Condition and Completeness of Eigenfunctions . . . .2199.2.4 Orthogonal Polynomials and Functions 220

9.3 The Hermite Polynomials 2239.4 The Helmholtz Differential Equation in Spherical Coordinates 225

9.4.1 Introduction 2259.4.2 Legendre Polynomials and Associated Legendre Functions 2279.4.3 Laguerre Polynomials and Associated Laguerre Polynomials 230

9.5 The Helmholtz Differential Equation in Cylindrical Coordinates 2339.5.1 Introduction 2339.5.2 Solutions of Bessel's Differential Equation 2339.5.3 Bessel Functions of the First Kind 2349.5.4 Neumann Functions 2349.5.5 Hankel Functions 2359.5.6 Modified Bessel Functions 2369.5.7 Spherical Bessel Functions 236

9.6 The Hypergeometric Function 2369.7 The Confluent Hypergeometric Function 2399.8 Other Special Functions used in Physics 240

9.8.1 Some Other Special Functions of Type 1 2409.8.2 Some Other Special Functions of Type 2 241

9.9 Problems 2429.9.1 Worksheet: The Quantum Mechanical Linear Harmonic Oscillator . . 2449.9.2 Worksheet: The Legendre Differential Equation 2479.9.3 Worksheet: The Laguerre Differential Equation 2499.9.4 Worksheet: The Bessel Differential Equation 2509.9.5 Worksheet: The Hypergeometric Differential Equation 251

CONTENTS 11

10 Integral Equations 26510.1 Introduction 26510.2 Integral Equations with Separable Kernels 26710.3 Integral Equations with Displacement Kernels 26910.4 The Neumann Series Method 26910.5 The Abel Problem 27010.6 Problems 272

11 Applied Functional Analysis 27511.1 Introduction 27511.2 Stationary Values of Certain Functions and Functionals 276

11.2.1 Maxima and Minima of Functions 27611.2.2 Method of Lagrange's Multipliers 27611.2.3 Maxima and Minima of a Certain Definite Integral 278

11.3 Hamilton's Variational Principle in Mechanics 28211.3.1 Introduction 28211.3.2 Generalized Coordinates 28211.3.3 Lagrange's Equations 28311.3.4 Format for Solving Problems by Use of Lagrange's Equations 284

11.4 Formulation of Hamiltonian Mechanics 28511.4.1 Derivation of Hamilton's Canonical Equations 28511.4.2 Format for Solving Problems by Use of Hamilton's Equations 28611.4.3 Poisson's Brackets 287

11.5 Continuous Media and Fields 28811.6 Transitions to Quantum Mechanics 288

11.6.1 Introduction 28811.6.2 The Heisenberg Picture 28911.6.3 The Schrodinger Picture 28911.6.4 The Feynman Path Integral 290

11.7 Problems 291

12 Geometrical Methods in Physics 29312.1 Introduction 29312.2 Transformation of Coordinates in Linear Spaces 29412.3 Contravariant and Covariant Tensors 296

12.3.1 Tensors of Rank One 29612.3.2 Higher-Rank Tensors 29712.3.3 Symmetric and Antisymmetric Tensors 29812.3.4 Polar and Axial Vectors 299

12.4 Tensor Algebra 29912.4.1 Addition (Subtraction) 29912.4.2 Multiplication (Outer Product) 29912.4.3 Contraction 30012.4.4 Inner Product 30012.4.5 The Quotient Law 300

12.5 The Line Element 30112.5.1 The Fundamental Metric Tensor 301

12 CONTENTS

12.5.2 Associate Tensors 30212.6 Tensor Calculus 302

12.6.1 Introduction 30212.6.2 Christoffel Symbols 30312.6.3 Covariant Differentiation of Tensors 304

12.7 The Equation of the Geodesic Line 30612.8 Special Equations Involving the Metric Tensor 307

12.8.1 The Riemann-Christoffel Tensor 30812.8.2 The Curvature Tensor 30912.8.3 The Ricci Tensor 30912.8.4 , The Einstein Tensor and Equations of General Relativity 309

12.9 Exterior Differential Forms 31012.9.1 Introduction 31012.9.2 Exterior Product 31212.9.3 Exterior Derivative 31312.9.4 The Exterior Product and Exterior Derivative in E3 31312.9.5 The Generalized Stokes Theorem 315

12.10Problems 315

Bibliography 317

Index 320