Shigeji Fujita and Salvador V Godoy. Mathematical Physics WILEY- VCH. WILEY-VCH Verlag GmbH & Co. KGaA

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1 Shigeji Fujita and Salvador V Godoy Mathematical Physics WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA

2 Contents Preface XIII Table of Contents and Categories XV Constants, Signs, Symbols, and General Remarks XVII 1 Vectors Definition and Important Properties Definitions Product of a Scalar and a Vector Position Vector Scalar Product Vector Product Differentiation Spherical Coordinates Cylindrical Coordinates 7 2 Tensors and Matrices Dyadic or Tensor Product Cartesian Representation Dot Product Unit Tensor Symmetric Tensor Eigenvalue Problem 15 3 Hamiltonian Mechanics Newtonian, Lagrangian and Hamiltonian Descriptions Newtonian Description Lagrangian Description Hamiltonian Description State of Motion in Phase Space. Reversible Motion Hamiltonian for a System of many Particles Canonical Transformation Poisson Brackets 36 Mathematical Physics. Shigeji Fujita and Salvador V. Godoy Copyright ( WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN:

3 VI I Contents 4 Coupled Oscillators and Normal Modes Oscillations of Particles an a String and Normal Modes Normal Coordinates 48 5 Stretched String Transverse Oscillations of a Stretched String Normal Coordinates for a String 58 6 Vector Calculus and the del Operator Differentiation in Time Space Derivatives The Gradient The Divergence The Curl Space Derivatives of Products Space Derivatives in Curvilinear Coordinates Spherical Coordinates (r, 0, yo) Cylindrical Coordinates (p,cp, z) Integral Theorems The Line Integral of VO Stokes's Theorem Gauss's Theorem Derivation of the Gradient, Divergence and Curl 73 7 Electromagnetic Waves Electric and Magnetic Fields in a Vacuum The Electromagnetic Field Theory 82 8 Fluid Dynamics Continuity Equation Fluid Equation of Motion Fluid Dynamics and Statistical Mechanics 92 9 Irreversible Processes Irreversible Phenomena, Viscous Flow, Diffusion Collision Rate and Mean Free Path Ohm's Law, Conductivity, and Matthiessen's Rule The Entropy Foundations of Thermodynamics The Carnot Cycle Carnot's Theorem Heat Engines and Refrigerating Machines Clausius's Theorem The Entropy The Exact Differential 122

4 Co ntents I VII 11 Thermodynamic Inequalities Irreversible Processes and the Entropy The Helmholtz Free Energy The Gibbs Free Energy Maxwell Relations Heat Capacities Nonnegative Heat Capacity and Compressibility Probability, Statistics and Density Probabilities Binomial Distribution Average and Root-Mean-Square Deviation. Random Walks Microscopic Number Density Dirac's Delta Function The Three-Dimensional Delta Function Liouville Equation Liouville's Theorem Probability Distribution Function. The Liouville Equation The Gibbs Ensemble Many Particles Moving in Three Dimensions More about the Liouville Equation Symmetries of Hamiltonians and Stationary States Generalized Vectors and Linear Operators Generalized Vectors. Matrices Linear Operators The Eigenvalue Problem Orthogonal Representation Quantum Mechanics for a Particle Quantum Description of a Linear Motion The Momentum Eigenvalue Problem The Energy Eigenvalue Problem Fourier Series and Transforms Fourier Series Fourier Transforms Bra and Ket Notations Heisenberg's Uncertainty Principle Quantum Angular Momentum Quantum Angular Momentum Properties of Angular Momentum Spin Angular Momentum The Spin Angular Momentum The Spin of the Electron 231

5 VIII 1 Contents The Magnetogyric Ratio 234 A. Free Electron 235 B. Free Proton 236 C. Free Neutron 237 D.Atomic Nuclei 237 E. Atoms and Ions 237 Time-Dependent Perturbation Theory 239 Perturbation Theory 1; The Dirac Picture 239 Scattering Problem; Fermi's Golden Rule 242 Perturbation Theory 2. Second Intermediate Picture 245 Laplace Transformation 249 Laplace Transformation 249 The Electric Circuit Equation 251 Convolution Theorem 252 Linear Operator Algebras 253 Quantum Harmonic Oscillator 255 Energy Eigenvalues 255 Quantum Harmonic Oscillator 259 Permutation Group 263 Permutation Group 263 Odd and Even Permutations 267 Quantum Statistics 273 Classical Indistinguishable Particles 273 Quantum-Statistical Postulate. Symmetric States for Bosons 276 Antisymmetric States for Fermions. Pauli's Exclusion Principle 278 Occupation-Number Representation 280 The Free-Electron Model 283 Free Electrons and the Fermi Energy 283 Density of States 287 Qualitative Discussion 291 Sommerfeld's Calculations 293 The Bose Einstein Condensation 297 Liquid Helium 297 The Bose Einstein Condensation of Free Bosons 298 Bosons in Condensed Phase 301 Magnetic Susceptibility 307 Introduction 307 Pauli Paramagnetism 308 Motion of a Charged Particle in Electromagnetic Fields 310 Electromagnetic Potentials 313 The Landau States and Energies 316

6 26.6 The Degeneracy of the Landau Levels Landau Diamagnetism Theory of Variations The Euler Lagrange Equation Fermat's Principle Hamilton's Principle Lagrange's Field Equation Second Quantization Boson Creation and Annihilation Operators Observables Fermions Creation and Annihilation Operators Heisenberg Equation of Motion Quantum Statistics of Composites Ehrenfest Oppenheimer Bethe's Rule Two-Particle Composites Discussion Superconductivity Basic Properties of a Superconductor Zero Resistance Meissner Effect Ring Supercurrent and Flux Quantization Josephson Effects Energy Gap Sharp Phase Change Occurrence of a Superconductor Elemental Superconductors Compound Superconductors High- Tc Superconductors Theoretical Survey The Cause of Superconductivity The Bardeen Cooper Schrieffer Theory Quantum-Statistical Theory The Full Hamiltonian Summary of the Results Complex Numbers and Taylor Series Complex Numbers Exponential and Logarithmic Functions Laws of Exponents Natural Logarithm Relationship between Exponential and Trigonometric Functions Hyperbolic Functions Definition of Hyperbolic Functions 378 Contents I IX

7 X I Contents Addition Formulas Double-Angle Formulas Sum, Difference and Product of Hyperbolic Functions Relationship between Hyperbolic and Trigonometric Functions Taylor Series Derivatives Taylor Series Binomial Series Series for Exponential and Logarithmic Functions Convergence of a Series Analyticity and Cauchy Riemann Equations The Analytic Function Poles Exponential Functions Branch Points Function with Continuous Singularities Cauchy Riemann Relations Cauchy Riemann Relations Applications Cauchy's Fundamental Theorem Cauchy's Fundamental Theorem Line Integrals Circular Integrals Cauchy's Integral Formula Laurent Series Taylor Series and Convergence Radius Uniform Convergence Laurent Series Multivalued Functions Square-Root Functions. Riemann Sheets and Cut Multivalued Functions Residue Theorem and Its Applications Residue Theorem Integrals ofthe Form fr x, dx f(x) Integrals ofthe Type f oz dxeix f( x ) 419 Integrals of the Type fö n d 0 f(cos 0,sin 0) Miscellaneous Integrals 421

8 Co ntents I XI Appendix A Representation-Independence of Poisson Brackets 423 Appendix B Proof of the Convolution Theorem 427 Appendix C Statistical Weight for the Landau States 431 Appendix D Useful Formulas 433 References 435 Index 439

Contents Preface Physical Constants, Units, Mathematical Signs and Symbols Introduction Kinetic Theory and the Boltzmann Equation

Contents Preface Physical Constants, Units, Mathematical Signs and Symbols Introduction Kinetic Theory and the Boltzmann Equation V Contents Preface XI Physical Constants, Units, Mathematical Signs and Symbols 1 Introduction 1 1.1 Carbon Nanotubes 1 1.2 Theoretical Background 4 1.2.1 Metals and Conduction Electrons 4 1.2.2 Quantum