The Mathematics of Classical and Quantum Physics

The Mathematics of Classical and Quantum Physics

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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."

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Product details

  • Paperback | 665 pages
  • 154.94 x 231.14 x 35.56mm | 839.14g
  • Dover Publications Inc.
  • New York, United States
  • English
  • New edition
  • New edition
  • Ill.
  • 048667164X
  • 9780486671642
  • 106,939

Table of contents

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

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