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# Mathematics of Classical and Quantum Physics

672
by Frederick W. Byron Jr., Robert W. FullerFrederick W. Byron Jr.

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## Overview

This textbook is designed to complement graduate-level physics texts in classical mechanics, electricity, magnetism, and quantum mechanics. Organized around the central concept of a vector space, the book includes numerous physical applications in the body of the text as well as many problems of a physical nature. It is also one of the purposes of this book to introduce the physicist to the language and style of mathematics as well as the content of those particular subjects with contemporary relevance in physics.

Chapters 1 and 2 are devoted to the mathematics of classical physics. Chapters 3, 4 and 5 — the backbone of the book — cover the theory of vector spaces. Chapter 6 covers analytic function theory. In chapters 7, 8, and 9 the authors take up several important techniques of theoretical physics — the Green's function method of solving differential and partial differential equations, and the theory of integral equations. Chapter 10 introduces the theory of groups. The authors have included a large selection of problems at the end of each chapter, some illustrating or extending mathematical points, others stressing physical application of techniques developed in the text.

Essentially self-contained, the book assumes only the standard undergraduate preparation in physics and mathematics, i.e. intermediate mechanics, electricity and magnetism, introductory quantum mechanics, advanced calculus and differential equations. The text may be easily adapted for a one-semester course at the graduate or advanced undergraduate level.

Chapters 1 and 2 are devoted to the mathematics of classical physics. Chapters 3, 4 and 5 — the backbone of the book — cover the theory of vector spaces. Chapter 6 covers analytic function theory. In chapters 7, 8, and 9 the authors take up several important techniques of theoretical physics — the Green's function method of solving differential and partial differential equations, and the theory of integral equations. Chapter 10 introduces the theory of groups. The authors have included a large selection of problems at the end of each chapter, some illustrating or extending mathematical points, others stressing physical application of techniques developed in the text.

Essentially self-contained, the book assumes only the standard undergraduate preparation in physics and mathematics, i.e. intermediate mechanics, electricity and magnetism, introductory quantum mechanics, advanced calculus and differential equations. The text may be easily adapted for a one-semester course at the graduate or advanced undergraduate level.

## Product Details

ISBN-13: | 9780486135069 |
---|---|

Publisher: | Dover Publications |

Publication date: | 03/29/2012 |

Series: | Dover Books on Physics |

Sold by: | Barnes & Noble |

Format: | NOOK Book |

Pages: | 672 |

Sales rank: | 1,114,867 |

File size: | 57 MB |

Note: | This product may take a few minutes to download. |

## Table of Contents

VOLUME ONE1 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

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