Introduction to Mathematical Physics

Title page
Preface
1 Vector Analysis
1.1 INTRODUCTION
1.1.1 Definition of Terms
1.1.2 Fundamental Concepts and Notations
1.1.3 Vector Addition without Coordinates
1.2 THE CARTESIAN SYSTEM OF BASE VECTORS
1.2.1 An Orthonrmal Basis
1.2.2 Position Vector (Radius Vector)
1.2.3 Rectangular Resolution of a Vector
1.2.4 Direction Cosines
1.2.5 Vector Algebra with Coordinates
1.3 DIFFERENTIATION OF VECTOR FUNCTIONS
1.3.1 The Derivative of a Vector
1.3.2 The Concept of a Gradient
1.4 INTEGRATION OF VECTOR FUNCTIONS
1.4.1 Line Integrals
1.4.2 The Divergence Theorem Due to Gauss
1.4.3 Green's Theorem
1.4.4 The Curl Theorem Due to Stokes
1.4.5 Two Useful Integral Relations
1.5 SOME USEFUL RELATIONS INVOLVING VECTORS
1.5.1 Relations Involving the ∇ Operator
I.5.2 Other Vector Relations
1.5.3 Some Important Equations in Physics
1.6 GENERALIZED COORDINATES
1.6,1 General Curvilinear Coordinates
1.6.2 Orthogonal Curvilinear Coordinates
1.6.3 The Gradient in Orthogonal Curvilinear Coordinates
1.6.4 The Divergence and Curl in Orthogonal Curvilinear Coordinates
1.6.5 The Laplacian in Orthogonal Curvilinear Coordinates
1.6.6 Plane Polar Coordinates
1.6.7 Right Circular Cylindrical Coordinates
1.6.8 Spherical Polar Coordinates
1.7 PROBLEMS
2 Operator and Matrix Analysis
2.1 INTRODUCTION
2.2 RUDIMENTS OF VECTOR SPACES
2.2.1 Definition of a Vector Space
2.2.2 Linear Dependence
2.2.3 Dimensionality of a Vector Space
2.2.4 Inner Product
2.2.5 Hilbert Space
2.2.6 Linear Operators
2.3 MATRIX ANALYSIS AND NOTATIONS
2.4 MATRIX OPERATIONS
2.4.1 Addition (Subtraction)
2.4.2 Multiplication
2.4.3 Division
2.4.4 The Derivative of tf Matrix
2.4.5 The Integral of a Matrix
2.4.6 Partitioned Matrices
2.5 PROPERTIES OF ARBITRARY MATRICES
2.5.1 Transpose Matrix
2.5.2 Complex Conjugate Matrix
2.5.3 Hermitian Conjugate
2.6 SPECIAL SQUARE MATRICES
2.6.1 Unit Matrix
2.6.2 Diagonal Matrix
2.6.3 Singular Matrix
2.6.4 Cofactor Matrix
2.6.5 Adjoint of a Matrix
2.6.6 Self-Adjoint Matrix
2.6.7 Symmetric Matrix
2.6.8 Antisymmetric (Skew) Matrix
2.6.9 Hermitian Matrix
2.6.10 Unitary Matrix
2.6.11 Orthogonal Matrix
2.6.12 The Trace of a Matrix
2.6.13 The Inverse Matrix
2.7 SOLUTION OF A SYSTEM OF LINEAR EQUATIONS
2.8 THE EIGENVALUE PROBLEM
2.9 COORDINATE TRANSFORMATIONS
2.9.1 Rotation in Two Dimensions
2.9.2 Rotation in Three Dimensions
2.10 PROBLEMS
APPENDIX: RUDIMENTS OF DETERMINANTS
3 Functions of a Complex Variable
3.1 INTRODUCTION
3.2 COMPLEX VARIABLES AND REPRESENTATIONS
3.2.1 Algebraic Operations
3.2.2 Argand Diagram: Vector Representation
3.2.3 Complex Conjugate
3.2.4 Euler's Formula
3.2.5 De Moivre's Theorem
3.2.6 The nth Root or Power of a Complex Number
3.3 ANALYTIC FUNCTIONS OF A COMPLEX VARIABLE
3.3.1 The Derivative of f(Z) and Analyticity
3.3.2 Harmonic Functions
3.3.3 Contour Integrals
3.3.4 Cauchy's Integral Theorem
3.3.5 Cauchy's Integral Formula
3.3.6 Differentiation Inside the Sign of Integration
3.4 SERIES EXPANSIONS
3.4.1 Taylor's Series
3.4.2 Laurent's Expansion
3.5 PROBLEMS
APPENDIX: RUDIMENTS OF SERIES
4 Calculus of Residues
4.1 ZEROS
4.2 ISOLATED SINGULAR POINTS
4.3 EVALUATION OF RESIDUES
4.3.1 mth-Order Pole
4.3.2 Simple Pole
4.4 THE CAUCHY RESIDUE THEOREM
4.5 THE CAUCHY PRINCIPAL VALUE
4.6 EVALUATION OF DEFINITE INTEGRALS
4.6.1 Integrals of the Form: etc
4.6.2 Integrals of the Form: etc
4.6.3 A Digression on Jordan's Lemma
4.6.4 Integrals of the Form: etc
4.7 DISPERSION RELATIONS
4.8 GEOMETRICAL REPRESENTATION
4.8.1 Introduction
4.8.2 Conformal Transformation (Mapping)
4.9 PROBLEMS
5 Differential Equations
5.1 INTRODUCTION
5.2 ORDINARY DIFFERENTIAL EQUATIONS
5.2.1 First-Order Homogeneous and Nonhomogeneous Equations with Variable Coefficients
5.2.2 The Superposition Principle
5.2.3 Second-Order Homogeneous Equations with Constant Coefficients
5.2.4 Second-Order Nonhomogeneous Equations with Constant Coefficients
5.2.5 Second-Order Nonhomogeneous Equations with Variable Coefficients
5.2.6 Second-Order Homogeneous Equations with Variable Coefficients
5.3 PARTIAL DIFFERENTIAL EQUATIONS
5.3.1 Introduction
5.3.2 Some Important Partial Differential Equations in Physics
5.3.3 An Illustration of the Method of Direct Integration
5.3.4 Method of Separation of Variables
5.4 PROBLEMS
6 Special Functions of Mathematical Physics
6.1 INTRODUCTION
6.2 THE HERMITE POLYNOMIALS
6.2.1 Basic Equations of Motion in Mechanics
6.2.2 The One-Dimensional Linear Harmonic Oscillator
6.2.3 The Solution of Hermite's Differentiai Equation
6.3 LEGENDRE AND ASSOCIATE LEGENDRE POLYNOMIALS
6.3.1 Spherical Harmonics
6.3.2 The Azimuthal Equation
6.3.3 Legendre Polynomials
6.4 THE CENTRAL FORCE PROBLEM
6.4.1 Introduction
6.4.2 The Laguerre Polynomials
6.4.3 Two Equations Reducible to Laguerre's Equation
6.5 BESSEL FUNCTIONS
6.5.1 Introduction
6.5.2 The Solution of Bessel's Equation
6.5.3 Analysis of the Various Solutions of Bessel's Equation
6.5.4 Neumann Functions
6.5.5 Hankel Functions
6.5.6 Modified Bessel Functions
6.5.7 Spherical Bessel Functions
6.5.8 The Characteristics of the Various Bessel Functions
6.5.9 Certain Other Special Functions
6.6 PROBLEMS
APPENDIX: THE RELATION BETWEEN P_l(W) AND P_l^m(W)
7 Fourier Series
7.1 INTRODUCTION
7.1.1 The Fourier Cosine and Sine Series
7.1.2 Change of Interval
7.1.3 Fourier Integral
7.1.4 Complex Form of the Fourier Series
7.2 GENERALIZED FOURIER SERIES AND THE DIRAC DELTA FUNCTION
7.3 SUMMATION OF THE FOURIER SERIES
7.4 THE GIBBS PHENOMENON
7.5 SUMMARY OF SOME PROPERTIES OF FOURIER SERIES
7.6 PROBLEMS
8 Fourier Transforms
8.1 INTRODUCTION
8.2 THEORY OF FOURIER TRANSFORMS
8.2.1 Formal Development of the Complex Fourier Transform
8.2.2 Cosine and Sine Transforms
8.2.3 Multiple-Dimensional Fourier Transforms
8.2.4 The Transforms of Derivatives
8.2.5 The Convolution Theorem
8.2.6 Parseval's Relation
8.3 THE WAVE PACKET IN QUANTUM MECHANICS
8.3.1 Origin of the Problem: Quantization of Energy
8.3.2 The Development of a New Quantum Theory
8.3.3 A Wave Equation for Particles: The Wave Packet
8.4 PROBLEMS
9 Tensor Analysis
9.1 INTRODUCTION
9.1.1 Notations
9.1.2 The Rank and Number of Components of a Tensor
9.2 TRANSFORMATION OF COORDINATES lN LINEAR SPACES
9.3 CONTRAVARIANT AND COVARIANT TENSORS
9.3.1 First-Rank Tensors (Vectors)
9.3.2 Higher-Rank Tensors
9.3.3 Symmetric and Antisymmetric Tensors
9.3.4 Polar and Axial Vectors
9.4 TENSOR ALGEBRA
9.4.1 Addition (Subtraction)
9.4.2 Multipfication (Outer Product)
9.4.3 Contraction
9.4.4 Inner Product
9.4.5 Quotient Law
9.5 THE LINE ELEMENT
9.5.1 The Fundamental Metric Tensor
9.5.2 Associate Tensors
9.6 TENSOR CALCULUS
9.6.1 Christoffel Symbols
9.6.2 Covariant Differentiation of Tensors
9.6.3 The Equation of the Geodetic Line
9.6.4 The Riemann-Christoffel Tensor
9.7 PROBLEMS
APPENDIX A: THE TRANSFORMATION LAWS FOR THE CHRISTOFFEL SYMBOLS
APPENDIX B: RUDIMENTS OF THE CALCULUS OF VARIATIONS
General Bibliography
References
Index