Some Mathematical Methods of Physics

by Gerald Goertzel, Nunzio Tralli

Overview

This well-rounded, thorough treatment for advanced undergraduates and graduate students introduces basic concepts of mathematical physics involved in the study of linear systems. The text emphasizes eigenvalues, eigenfunctions, and Green's functions. Prerequisites include differential equations and a first course in theoretical physics.
The three-part presentation begins with an exploration of systems with a finite number of degrees of freedom (described by matrices). In part two, the concepts developed for discrete systems in previous chapters are extended to continuous systems. New concepts useful in the treatment of continuous systems are also introduced. The final part examines approximation methods — including perturbation theory, variational methods, and numerical methods — relevant to addressing most of the problems of nature that confront applied physicists. Two Appendixes include background and supplementary material. 1960 edition.

Product Details

ISBN-13:	9780486793320
Publisher:	Dover Publications
Publication date:	02/08/2014
Series:	Dover Books on Physics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	320
File size:	24 MB
Note:	This product may take a few minutes to download.

About the Author

American theoretical physicist Gerald Goertzel (1919–2002) worked on the Manhattan Project, formed Sage Industries, and worked for many years at IBM, where he was active in design automation, data compression, and digital printing. He developed the Goertzel algorithm, a data signal processing technique.
Nunzio Tralli (1917–79) taught at St. John's University and was the author of several books on physics, including Classical Electromagnetic Theory and Atomic Theory: An Introduction to Wave Mechanics, which he co-wrote with Frank R. Pomilla.

Table of Contents

Preface v

References xiii

Part 1 Systems With A Finite Number Of Degrees Of Freedom

Chapter 1 Formulation of the Problem and Development of Notation 3

1.1 Introduction 3

1.2 Standardization of Notation 5

1.3 Matrices 7

1.4 Elementary Arithmetic Operations with Matrices 9

1.5 The Row-Column Rule 12

1.6 Warnings 13

1.7 Some Properties of Determinants 14

1.8 Inverses 16

1.9 Linear Independence 17

Chapter 2 Solution for Diagonalizable Matrices 21

2.1 Solution by Taylor Series 21

2.2 Eigenvalues and Eigencolumns 22

2.3 Superposition 23

2.4 Completeness 24

2.5 Diagonalization of Nondegenerate Matrices 27

2.6 Outline of Computation Procedure with Examples 29

2.7 Change of Variable 32

2.8 The Steady-state Solution 34

2.9 The Inhomogeneous Problem 35

Chapter 3 The Evaluation of a Function of a Matrix for an Arbitrary Matrix 38

3.1 Introduction 38

3.2 The Cauchy-integral Formula 38

3.3 Application to Matrices 39

3.4 Evaluation of f(A) with Illustrations 40

3.5 The Inversion Formula 43

3.6 Laplace Transforms 44

3.7 Inhomogeneous Equations 46

3.8 The Convolution Theorem 47

Chapter 4 Vector Spaces and Linear Operators 50

4.1 Introduction 50

4.2 Base Vectors and Basis 52

4.3 Change of Basis 55

4.4 Linear Operators 57

4.5 The Representation of Linear Operators by Matrices 57

4.6 The Operator in the Dual Space 58

14.7 Effect of Change of Basis on the Representation of an Operator 59

4.8 The Spectral Representation of an Operator 60

4.9 The Formation of a Basis by Eigenvectors of a Linear Operator 61

4.10 The Diagonalization of Normal Matrices 64

Chapters 5 The Dirac Notation 67

5.1 Introduction 67

5.2 The Change of Basis 68

5.3 Linear Operators in the Dirac Notation 69

5.4 Eigenvectors and Eigenvalues 70

5.5 The Spectral Representation of an Operator 70

5.6 Theorems on Hermitian Operators 70

Chapter 6 Periodic Structures 73

6.1 Motivation 73

6.2 The RC Line 73

6.3 Diagonalizing M 75

6.4 The Loaded String 78

6.5 Difference Operators 79

Part 2 Systems With An Infinite Number Of Degrees Of Freedom

Chapter 7 The Transition to Continuous Systems 85

7.1 Introduction 85

7.2 The RC Line?Change of Notation 85

7.3 The RC Line?Transition to the Continuous Case 87

7.4 Solution of the Discrete Problem 87

7.5 Solution in the Limit (Continuous Problem) 89

7.6 The Fourier Transform 92

Chapter 8 Operators in Costinuous Systems 93

8.1 Introduction 96

8.2 Operators on Functions 97

8.3 The Dirac δ Function 98

8.4 Coordinate Transformations 100

8.5 Adjoints 103

8.6 Orthogonality of Eigenfunctions 104

8.7 Functions of Operators 105

8.8 Three-dimensional Continuous Systems 107

8.9 Differential Operators 108

Chapter 9 Hie Lapladan (∇2) in One Dimension 111

9.1 Introduction 111

9.2 The Infinite Domain, ? ∞ <x <+ ∞ 112

9.3 The Semi-infinite Domain, 0 ≤ x < + ∞ 116

9.4 The Finite Domain, 0 ≤ x ≤ L 119

9.5 The Circular Domain 122

9.6 The Method of Images 123

Chapter 10 The Lapiacian (∇2) in Two Dimensions 128

10.1 Introduction 128

10.2 Conduction of Heat in an Infinite Insulated PlateCartesian Coordinates 128

10.3 The Vibrating Rectangular Membrane 130

10.4 Conduction of Heat in an Infinite Insulated Plate; Plane Polar Coordinates 131

10.5 Theorems on Cylindrical Functions (of Integral Order n) 134

10.6 Conduction of Heat in an Infinite Insulated Plate; Plane Polar Coordinates (Concluded) 140

10.7 The Circular Membrane 141

10.8 The Vibrating Circular Ring and Circular Sector 143

Chapter 11 The Lapiacian (∇2) in Three Dimensions 146

11.1 Introduction 146

11.2 The Wave Equation in Three Dimensions 147

11.3 The Eigenvalues of L2 andLx 149

11.4 The Simultaneous Eigenfunctions of L and Lz 152

11.5 Solution of ∇² φ = 0 154

11.6 Solution of (∇² + k²)φ 155

11.7 Recurrence Relations for the Spherical Harmonics 158

11.8 Some Expansion Theorems 160

11.9 Solution of the Wave Equation 162

11.10 Heat Conduction in an Infinite Solid 163

Chapter 12 Green's Functions 165

12.1 Definition 165

12.2 The Necessary and Sufficient Condition for a Green's Function 166

12.3 The Operator -α²d²/dx² + 1 in an Infinite Domain 167

12.4 The Operator ?α²d²/dx² + 1 in a Finite Domain 169

12.5 The Operator ?α²∇² + 1 in Spherical Coordinates 171

Chapter 13 Radiation and Scattering Problems 176

13.1 The Outgoing Wave Condition 176

13.2 The Green's Function Solution 177

13.3 The Multipole Expansion 179

13.4 The Radiation Far from the Source 180

13.5 Radiation from an Infinitely Long Cylinder 181

13.6 The Scattering Problem 183

13.7 The Scattering Cross Section 184

13.8 The Method of Partial Waves 185

13.9 The Born Approximation 190

13.30 Gratings 192

Part 3 approximate Methods

Chapter 14 Perturbation of Eigenvalues 201

14.1 Introduction 201

14.2 Formulation of the Problem 202

14.3 A Simple Solution 203

14.4 Nondegenerate Eigenvalues 204

14.5 Change of Notation and an Extension 205

14.6 An Application?The Vibrating String 206

14.7 Degenerate Eigenvalues 207

Chapter 15 Variational Estimates 211

15.1 Introduction 211

15.2 The Rayleigh Variational Principle 212

15.3 A Lower Bound 213

15.4 The Ritz Method 215

15.5 Higher Eigenvalues by the Ritz Method 220

15.6 Example of the Ritz Method 221

Chapter 16 Iteration Procedures 225

16.1 Introduction 225

16.2 Eigenvalue Problems 225

16.3 Inverses by Iteration 229

Chapter 17 Construction of Eigenvalue Problems 233

17.1 Introduction 233

17.2 The Method 233

17.3 Application to the Scattering Problem 234

Chapter 18 Numerical Procedures 236

18.1 Introduction 236

18.2 Simplification of the Model 236

18.3 Difference Equations from the Variational Principle 239

Appendix 1A Determinants 243

Appendix IB Convergence of Matrix Power Series 252

Appendix 1C Remarks on Theory of Functions of Complex Variables 254

1C.I Analytic Functions 254

1C.2 The Cauchy Integral Theorem and Corollary 255

1C.3 Singularities 256

1C.4 Cauchy's Integral Formula 257

1C.5 The Theorem of Residues 259

Appendix 2A Evaluation of Integrals of the Form$$$ 261

Appendix 2B Fourier Transforms, Integrals, and Series 266

2B.1 Introduction 266

2B.2 Transforms 267

2B.3 Infinite One-dimensional Transforms 268

2B.4 Infinite Multidimensional Transforms?Cartesian Coordinates 271

2B.5 Finite One-dimensional Transforms 272

2B.6 The Fourier-Bessel Integral 275

2B.7 The Fourier-Bessel Expansion 276

Appendix 2C The Cylindrical Functions 280

2C.1 Introduction 280

2C.2 The Integral Representation of J_n(p) 280

2C.3 The Integral Representations of the General Cylindrical Functions 281

2C.4 The Integral Representation of the Bessel Function J_v 283

2C.5 The Hankel Functions 284

2C.6 Series Expansions at the Origin 286

2C.7 The Asymptotic Expansions 287

2C.8 The Asymptotic Series of Debye 291

2C.9 The Addition Theorems for Bessel Functions 292

Index 295