Book:Frederick W. Byron, Jr./Mathematics of Classical and Quantum Physics

A republication of and  in one volume.

Subject Matter

 * Physics
 * Quantum Mechanics

Contents

 * Preface


 * 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 Matrices
 * Introduction
 * 3.1 Groups, Fields, and Vector Spaces
 * 3.2 Linear Independence
 * 3.3 Bases and Dimensionality
 * 3.4 Isomorphisms
 * 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 Orthonormal 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 Perturbation Theory - Nondegenerate Case
 * *4.12 Perturbation 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 $\delta$-Function
 * 5.4 Weierstrass's Theorem: Approximation by Polynomials
 * 5.5 Legendre Polynomials
 * 5.6 Fourier Series
 * 5.7 Fourier Integrals
 * 5.8 Spherical Harmonics and Associated Legendre Functions
 * 5.9 Hermite Polynomials
 * 5.10 Sturm-Liouville Systems - Orthogonal 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 Functions
 * 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 Perturbation Theory
 * 9.3 Finite-Rank Techniques for Eigenvalue Problems
 * 9.4 The Fredholm Alternative For Completely Continuous Operators
 * 9.5 The Numerical Solution 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



Dot Product: Einstein Summation Convention
Volume One: Chapter $1$ Vectors in Classical Physics: $1.3$ The Scalar Product:

Source work progress
* : Volume One: Chapter $1$ Vectors in Classical Physics: $1.3$ The Scalar Product