Mathematical Physics

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List of Published Books

Mathematical Physics

© 2010 WILEY-VCH

     ISBN 978-3-527-40808-5

Contents

Preface

Table of Contents and Categories

Constants, Signs, Symbols, and General Remarks

List of Symbols

1  Vectors

     1.1  Definition and Important Properties

          1.1.1 Definitions

     1.2 Product of a Scalar and a Vector

     1.3 Position Vector

     1.4 Scalar Product

     1.5 Vector Product

     1.6 Differentiation

     1.7 Spherical Coordinates

     1.8 Cylindrical Coordinates

2  Tensors and Matrices

     2.1 Dyadic or Tensor Product

     2.2 Cartesian Representation

     2.3 Dot Product

          2.3.1 Unit Tensor

     2.4 Symmetric Tensor

     2.5 Eigenvalue Problem

3  Hamiltonian Mechanics

     3.1 Newtonian, Lagrangian and Hamiltonian Descriptions

          3.1.1 Newtonian Description

          3.1.2 Lagrangian Description

          3.1.3 Hamiltonian Description

     3.2 State of Motion in Phase Space. Reversible Motion

     3.3 Hamiltonian for a System of many Particles

     3.4 Canonical Transformation

     3.5 Poisson Brackets

     References

4  Coupled Oscillators and Normal Modes

     4.1 Oscillations of Particles on a String and Normal Modes

     4.2 Normal Coordinates

5  Stretched String

     5.1 Transverse Oscillations of a Stretched String

     5.2 Normal Coordinates for a String

6  Vector Calculus and the del Operator

     6.1 Differentiation in Time

     6.2 Space Derivatives

          6.2.1 The Gradient

          6.2.2 The Divergence

          6.2.3 The Curl

          6.2.4 Space Derivatives of Products

     6.3 Space Derivatives in Curvilinear Coordinates

          6.3.1 Spherical Coordinates (r, \theta, \phi)

          6.3.2 Cylindrical Coordinates

     6.4 Integral Theorems

          6.4.1The Line Integral of \Grad \phi

          6.4.2 Stokes's Theorem

     6.5 Gauss's Theorem

     6.6 Derivation of the Gradient, Divergence and Curl

7  Electromagnetic Waves

     7.1 Electric and Magnetic Fields in a Vacuum

     7.2 The Electromagnetic Field Theory

8  Fluid Dynamics

     8.1 Continuity Equation

     8.2 Fluid Equation of Motion

     8.3 Fluid Dynamics and Statistical Mechanics

9  Irreversible Processes

     9.1 Irreversible Phenomena, Viscous Flow, Diffusion

     9.2 Collision Rate and Mean Free Path

     9.3 Ohm's Law, Conductivity, and Matthiessen's Rule

10  The Entropy

     10.1 Foundations of Thermodynamics

     10.2 The Carnot Cycle

     10.3 Carnot's Theorem

     10.4 Heat Engines and Refrigerating Machines

     10.5 Clausius's Theorem

     10.6 The Entropy

     10.7 The Exact Differential

     Reference

11  Thermodynamic Inequalities

     11.1 Irreversible Processes and the Entropy

     11.2 The Helmholtz Free Energy

     11.3 The Gibbs Free Energy

     11.4 Maxwell Relations

     11.5 Heat Capacities

     11.6 Nonnegative Heat Capacity and Compressibility

     References

12  Probability, Statistics and Density

     12.1 Probabilities

     12.2 Binomial Distribution

     12.3 Average and Root-Mean-Square Deviation. Random Walks

     12.4 Microscopic Number Density

     12.5 Dirac's Delta Function

     12.6 The Three-Dimensional Delta Function

13  Liouville Equation

     13.1 Liouville's Theorem

     13.2 Probability Distribution Function. The Liouville Equation

     13.3 The Gibbs Ensemble

     13.4 Many Particles Moving in Three Dimensions

     13.5 More about the Liouville Equation

     13.6 Symmetries of Hamiltonians and Stationary States

14  Generalized Vectors and Linear Operators

     14.1 Generalized Vectors. Matrices

     14.2 Linear Operators

     14.3 The Eigenvalue Problem

     14.4 Orthogonal Representation

15  Quantum Mechanics for a Particle

     15.1 Quantum Description of a Linear Motion

     15.2 The Momentum Eigenvalue Problem

     15.3 The Energy Eigenvalue Problem

16  Fourier Series and Transforms

     16.1 Fourier Series

     16.2 Fourier Transforms

     16.3 Bra and Ket Notations

     16.4 Heisenberg's Uncertainty Principle

17  Quantum Angular Momentum

     17.1 Quantum Angular Momentum

     17.2 Properties of Angular Momentum

18  Spin Angular Momentum

     18.1 The Spin Angular Momentum

     18.2 The Spin of the Electron

     18.3 The Magnetogyric Ratio

          18.3.1 A. Free Electron

          18.3.2 B. Free Proton

          18.3.3 C. Free Neutron

          18.3.4 D. Atomic Nuclei

          18.3.5 E. Atoms and Ions

19  Time-Dependent Perturbation Theory

     19.1 Perturbation Theory 1; The Dirac Picture

     19.2 Scattering Problem; Fermi's Golden Rule

     19.3 Perturbation Theory 2. Second Intermediate Picture

     References

20  Laplace Transformation

     20.1 Laplace Transformation

     20.2 The Electric Circuit Equation

     20.3 Convolution Theorem

     20.4 Linear Operator Algebras

21  Quantum Harmonic Oscillator

     21.1 Energy Eigenvalues

     21.2 Quantum Harmonic Oscillator

     Reference

22  Permutation Group

     22.1 Permutation Group

     22.2 Odd and Even Permutations

23  Quantum Statistics

     23.1 Classical Indistinguishable Particles

     23.2 Quantum-Statistical Postulate. Symmetric States for Bosons

     23.3 Antisymmetric States for Fermions. Pauli's Exclusion Principle

     23.4 Occupation-Number Representation

24  The Free-Electron Model

     24.1 Free Electrons and the Fermi Energy

     24.2 Density of States

     24.3 Qualitative Discussion

     24.4 Sommerfeld's Calculations

25  The Bose-Einstein Condensation

     25.1 Liquid Helium

     25.2 The Bose-Einstein Condensation of Free Bosons

     25.3 Bosons in Condensed Phase

     References

26  Magnetic Susceptibility

     26.1 Introduction

     26.2 Pauli Paramagnetism

     26.3 Motion of a Charged Particle in Electromagnetic Fields

     26.4 Electromagnetic Potentials

     26.5 The Landau States and Energies

     26.6 The Degeneracy of the Landau Levels

     26.7 Landau Diamagnetism

     References

27  Theory of Variations

     27.1 The Euler-Lagrange Equation

     27.2 Fermat's Principle

     27.3 Hamilton's Principle

     27.4 Lagrange's Field Equation

28  Second Quantization

     28.1 Boson Creation and Annihilation Operators

     28.2 Observables

     28.3 Fermions Creation and Annihilation Operators

     28.4 Heisenberg Equation of Motion

     Reference

29  Quantum Statistics of Composites

     29.1 Ehrenfest-Oppenheimer-Bethe's Rule

     29.2 Two-Particle Composites

     29.3 Discussion

     References

30  Superconductivity

     30.1 Basic Properties of a Superconductor

          30.1.1 Zero Resistance

          30.1.2 Meissner Effect

          30.1.3 Ring Supercurrent and Flux Quantization

          30.1.4 Josephson Effects

          30.1.5 Energy Gap

          30.1.6 Sharp Phase Change

     30.2 Occurrence of a Superconductor

          30.2.1 Elemental Superconductors

          30.2.2 Compound Superconductors

          30.2.3 High-T, Superconductors

     30.3 Theoretical Survey

          30.3.1 The Cause of Superconductivity

          30.3.2 The Bardeen-Cooper-Schrieffer Theory

     30.4 Quantum-Statistical Theory

          30.4.1 The Full Hamiltonian

          30.4.2 Summary of the Results

     References

31  Complex Numbers and Taylor Series

     31.1 Complex Numbers

     31.2 Exponential and Logarithmic Functions

          31.2.1 Laws of Exponents

          31.2.2 Natural Logarithm

          31.2.3 Relationship between Exponential and Trigonometric Functions

     31.3 Hyperbolic Functions

          31.3.1 Definition of Hyperbolic Functions

          31.3.2 Addition Formulas

          31.3.3 Double-Angle Formulas

          31.3.4 Sum, Difference and Product of Hyperbolic Functions

          31.3.5 Relationship between Hyperbolic and Trigonometric Functions

     31.4 Taylor Series

          31.4.1 Derivatives

          31.4.2 Taylor Series

          31.4.3 Binomial Series

          31.4.4 Series for Exponential and Logarithmic Functions

     31.5 Convergence of a Series

32  Analyticity and Cauchy-Riemann Equations

     32.1 The Analytic Function

     32.2 Poles

     32.3 Exponential Functions

     32.4 Branch Points

     32.5 Function with Continuous Singularities

     32.6 Cauchy-Riemann Relations

     32.7 Cauchy-Riemann Relations Applications

33  Cauchy's Fundamental Theorem

     33.1 Cauchy's Fundamental Theorem

     33.2 Line Integrals

     33.3 Circular Integrals

     33.4 Cauchy's Integral Formula

34  Laurent Series

     34.1 Taylor Series and Convergence Radius

     34.2 Uniform Convergence

     34.3 Laurent Series

35  Multivalued Functions

     35.1 Square-Root Functions. Riemann Sheets and Cut

     35.2 Multivalued Functions

36  Residue Theorem and Its Applications

     36.1 Residue Theorem

     36.2 Integrals of the Form

     36.3 Integrals of the Type

     36.4 Integrals of the Type Int(f(cos\theta), sin\theta),, 0, 2\pi)

     36.5 Miscellaneous Integrals

Appendix A  Representation-Independence of Poisson Brackets

Appendix B  Proof of the Convolution Theorem

Appendix C  Statistical Weight for the Landau States

Appendix D  Useful Formulas

References

Index