Mathematics of Classical and Quantum Physics

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