Mathematics of Classical and Quantum Physics

Well-organized text designed to complement graduate-level physics texts in classical mechanics, electricity, magnetism, and quantum mechanics. Topics include theory of vector spaces, analytic function theory, Green's function method of solving differential and partial differential equations, theory of groups, more. Many problems, suggestions for further reading.

VOLUME ONE
1 Vectors in Classical Physics
    Introduction
    1.1 Geometric and Algebraic Definitions of a Vector
    1.2 The Resolution of a Vector into Components
    1.3 The Scalar Product
    1.4 Rotation of the Coordinate System: Orthogonal Transformations
    1.5 The Vector Product
    1.6 A Vector Treatment of Classical Orbit Theory
    1.7 Differential Operations on Scalar and Vector Fields
    *1.8 Cartesian-Tensors
2 Calculus of Variations
    Introduction
    2.1 Some Famous Problems
    2.2 The Euler-Lagrange Equation
    2.3 Some Famous Solutions
    2.4 Isoperimetric Problems - Constraints
    2.5 Application to Classical Mechanics
    2.6 Extremization of Multiple Integrals
    2.7 Invariance Principles and Noether's Theorem
3 Vectors and Matrics
    Introduction
    3.1 "Groups, Fields, and Vector Spaces"
    3.2 Linear Independence
    3.3 Bases and Dimensionality
    3.4 Ismorphisms
    3.5 Linear Transformations
    3.6 The Inverse of a Linear Transformation
    3.7 Matrices
    3.8 Determinants
    3.9 Similarity Transformations
    3.10 Eigenvalues and Eigenvectors
    *3.11 The Kronecker Product
4. Vector Spaces in Physics
    Introduction
    4.1 The Inner Product
    4.2 Orthogonality and Completeness
    4.3 Complete Ortonormal Sets
    4.4 Self-Adjoint (Hermitian and Symmetric) Transformations
    4.5 Isometries-Unitary and Orthogonal Transformations
    4.6 The Eigenvalues and Eigenvectors of Self-Adjoint and Isometric Transformations
    4.7 Diagonalization
    4.8 On The Solvability of Linear Equations
    4.9 Minimum Principles
    4.10 Normal Modes
    4.11 Peturbation Theory-Nondegenerate Case
    4.12 Peturbation Theory-Degenerate Case
5. Hilbert Space-Complete Orthonormal Sets of Functions
    Introduction
    5.1 Function Space and Hilbert Space
    5.2 Complete Orthonormal Sets of Functions
    5.3 The Dirac d-Function
    5.4 Weirstrass's Theorem: Approximation by Polynomials
    5.5 Legendre Polynomials
    5.6 Fourier Series
    5.7 Fourier Integrals
    5.8 Sphereical Harmonics and Associated Legendre Functions
    5.9 Hermite Polynomials
    5.10 Sturm-Liouville Systems-Orthogaonal Polynomials
    5.11 A Mathematical Formulation of Quantum Mechanics
VOLUME TWO
6 Elements and Applications of the Theory of Analytic Functions
    Introduction
    6.1 Analytic Functions-The Cauchy-Riemann Conditions
    6.2 Some Basic Analytic Functions
    6.3 Complex Integration-The Cauchy-Goursat Theorem
    6.4 Consequences of Cauchy's Theorem
    6.5 Hilbert Transforms and the Cauchy Principal Value
    6.6 An Introduction to Dispersion Relations
    6.7 The Expansion of an Analytic Function in a Power Series
    6.8 Residue Theory-Evaluation of Real Definite Integrals and Summation of Series
    6.9 Applications to Special Functions and Integral Representations
7 Green's Function
    Introduction
    7.1 A New Way to Solve Differential Equations
    7.2 Green's Functions and Delta Functions
    7.3 Green's Functions in One Dimension
    7.4 Green's Functions in Three Dimensions
    7.5 Radial Green's Functions
    7.6 An Application to the Theory of Diffraction
    7.7 Time-dependent Green's Functions: First Order
    7.8 The Wave Equation
8 Introduction to Integral Equations
    Introduction
    8.1 Iterative Techniques-Linear Integral Operators
    8.2 Norms of Operators
    8.3 Iterative Techniques in a Banach Space
    8.4 Iterative Techniques for Nonlinear Equations
    8.5 Separable Kernels
    8.6 General Kernels of Finite Rank
    8.7 Completely Continuous Operators
9 Integral Equations in Hilbert Space
    Introduction
    9.1 Completely Continuous Hermitian Operators
    9.2 Linear Equations and Peturbation Theory
    9.3 Finite-Rank Techniques for Eigenvalue Problems
    9.4 the Fredholm Alternative for Completely Continuous Operators
    9.5 The Numerical Solutions of Linear Equations
    9.6 Unitary Transformations
10 Introduction to Group Theory
    Introduction
    10.1 An Inductive Approach
    10.2 The Symmetric Groups
    10.3 "Cosets, Classes, and Invariant Subgroups"
    10.4 Symmetry and Group Representations
    10.5 Irreducible Representations
    10.6 "Unitary Representations, Schur's Lemmas, and Orthogonality Relations"
    10.7 The Determination of Group Representations
    10.8 Group Theory in Physical Problems
General Bibliography
Index to Volume One
Index to Volume Two