Theoretical Physics Reference

• 1. Introduction
  • 1.1. Preface
  • 1.2. Introduction
• 2. Contributors
• 3. Mathematics
  • 3.1. Integration
    • 3.1.1. General Case
    • 3.1.2. Special Cases
    • 3.1.3. Motivation
    • 3.1.4. Example
  • 3.2. Complex Numbers
    • 3.2.1. Real and Imaginary Part
    • 3.2.2. Argument Function
    • 3.2.3. Exponential
    • 3.2.4. Logarithm
    • 3.2.5. Power
    • 3.2.6. Examples
    • 3.2.7. Complex Conjugate
    • 3.2.8. Complex Derivatives
    • 3.2.9. Testing Identities Using Computer Code
  • 3.3. Residue Theorem
    • 3.3.1. Computation of Residues
    • 3.3.2. Useful Formulas
    • 3.3.3. Complex Substitution
  • 3.4. Fourier Transform
    • 3.4.1. Shift Theorem
    • 3.4.2. Scaling
    • 3.4.3. Derivative
    • 3.4.4. Convolution
    • 3.4.5. Radial Fourier Transform
    • 3.4.6. Examples
  • 3.5. Fourier Transform of a Periodic Function (e.g. in a Crystal)
    • 3.5.1. One Dimension (Fourier Series)
  • 3.6. Discrete Fourier Transform
  • 3.7. Fast Fourier Transform (FFT)
    • 3.7.1. Decimation In Frequency (DIF)
  • 3.8. Laplace Transform
  • 3.9. Hilbert Transform
  • 3.10. Periodic Functions
  • 3.11. Polar Coordinates
    • 3.11.1. Example
  • 3.12. Spherical Coordinates
    • 3.12.1. Example I
    • 3.12.2. Example II
  • 3.13. Argument function, atan2
  • 3.14. Multiple Argument Formulas
    • 3.14.1. sin(a x)
    • 3.14.2. cos(a x)
  • 3.15. Delta Function
  • 3.16. Distributions
  • 3.17. Variations and Functional Derivatives
    • 3.17.1. Functions of One Variable
    • 3.17.2. Functions of several variables
    • 3.17.3. Functionals
    • 3.17.4. Examples
  • 3.18. Dirac Notation
  • 3.19. Homogeneous Functions (Euler’s Theorem)
    • 3.19.1. Example 1
    • 3.19.2. Example 2
  • 3.20. Green Functions
    • 3.20.1. Introduction
    • 3.20.2. Boundary Conditions
    • 3.20.3. Symmetry
    • 3.20.4. Examples
  • 3.21. Binomial Coefficients
  • 3.22. Double Sums
  • 3.23. Triangle Inequality
  • 3.24. Gamma Function
  • 3.25. Incomplete Gamma Function
    • 3.25.1. Example
  • 3.26. Factorial
  • 3.27. Double Factorial
    • 3.27.1. Example
  • 3.28. Fermi-Dirac Integral
  • 3.29. Legendre Polynomials
    • 3.29.1. Example I
    • 3.29.2. Example II
    • 3.29.3. Example III
    • 3.29.4. Example IV
  • 3.30. Spherical Harmonics
    • 3.30.1. Examples
  • 3.31. Gaunt Coefficients
    • 3.31.1. Example I
    • 3.31.2. Example II
    • 3.31.3. Example III
    • 3.31.4. Example IV
    • 3.31.5. Example V
    • 3.31.6. Example VI
  • 3.32. Wigner 3j Symbols
    • 3.32.1. Examples
  • 3.33. Multipole Expansion
  • 3.34. Hypergeometric Functions
    • 3.34.1. Convergence Conditions
    • 3.34.2. Elementary and Special Functions
    • 3.34.3. Example I
    • 3.34.4. Example II
    • 3.34.5. Example III
  • 3.35. Feynman Parameters
    • 3.35.1. Example 1
    • 3.35.2. Example 2
  • 3.36. Groups
    • 3.36.1. Theory
    • 3.36.2. Crystallographic Point Groups
    • 3.36.3. Applications of finite groups
    • 3.36.4. Continuous Groups
    • 3.36.5. Literature
  • 3.37. Wigner D Function
    • 3.37.1. Derivation
  • 3.38. Ordinary Differential Equations
    • 3.38.1. Finite Difference Formulas
    • 3.38.2. Integrating ODE
    • 3.38.3. Radial Poisson Equation
  • 3.39. Linear Algebra
    • 3.39.1. Scalar Product
    • 3.39.2. Projections
  • 3.40. Differential Geometry
    • 3.40.1. Manifolds
    • 3.40.2. Examples
  • 3.41. Operators
    • 3.41.1. Introduction
    • 3.41.2. Spectrum
    • 3.41.3. Derivative Operator
    • 3.41.4. Sturm–Liouville Operator
    • 3.41.5. Angular Momentum Operator
  • 3.42. Variational Formulation of PDEs
    • 3.42.1. Poisson Equation
    • 3.42.2. Radial Schrödinger Equation
• 4. Classical Mechanics, Special and General Relativity
  • 4.1. Gravitation and Electromagnetism as a Field Theory
    • 4.1.1. Gravitation
    • 4.1.2. Electromagnetism
    • 4.1.3. Relativistic Dust
    • 4.1.4. Equations of Motion
  • 4.2. Classical Mechanics
    • 4.2.1. Rigid Body Rotation
  • 4.3. Relativity
    • 4.3.1. Introduction: Why Tensors
    • 4.3.2. High School Formulation
    • 4.3.3. College Formulation
    • 4.3.4. Differential Geometry Formulation
    • 4.3.5. Metrics
    • 4.3.6. Conclusion About Metric
    • 4.3.7. Einstein’s Equations
    • 4.3.8. Continuous Distribution of Matter
    • 4.3.9. Obsolete Section
    • 4.3.10. Inertial frames
    • 4.3.11. Lorentz Group
    • 4.3.12. O(4) Group
    • 4.3.13. Proper Time
    • 4.3.14. FAQ
    • 4.3.15. Questions Without Answers (Yet)
• 5. Classical Electromagnetism
  • 5.1. Maxwell’s Equations
    • 5.1.1. Electromagnetic Field
    • 5.1.2. Maxwell’s Equations
    • 5.1.3. Lorentz Force
    • 5.1.4. Hamiltonian
    • 5.1.5. Electromagnetic Stress Tensor
    • 5.1.6. Examples
  • 5.2. Semiconductor Device Physics
    • 5.2.1. Equations
    • 5.2.2. Example 3
• 6. Thermodynamics and Statistical Physics
  • 6.1. Thermodynamics
    • 6.1.1. Fundamental Thermodynamic Relation
    • 6.1.2. Thermodynamic Potentials
    • 6.1.3. Examples
  • 6.2. Statistical Physics
    • 6.2.1. Microcanonical Ensemble
    • 6.2.2. Canonical Ensemble
    • 6.2.3. Grand Canonical Ensemble
    • 6.2.4. Examples
• 7. Fluid Dynamics
  • 7.1. Fluid Dynamics
    • 7.1.1. Stress-Energy Tensor
    • 7.1.2. Navier-Stokes Equations
    • 7.1.3. Incompressible Equations
    • 7.1.4. Bernoulli’s Principle
  • 7.2. MHD Equations
    • 7.2.1. Introduction
    • 7.2.2. Derivation
    • 7.2.3. Finite Element Formulation
  • 7.3. Compressible Euler Equations
    • 7.3.1. Introduction
    • 7.3.2. Dimensionless Euler Equations
    • 7.3.3. Conservative Form of the Euler Equations
    • 7.3.4. Weak Formulation
    • 7.3.5. Flux Jacobians
    • 7.3.6. 2D Version of the Equations
    • 7.3.7. Sea Breeze Modeling
    • 7.3.8. Boundary Conditions
    • 7.3.9. Newton Method
    • 7.3.10. Older notes
• 8. Quantum Field Theory and Quantum Mechanics
  • 8.1. Introduction
  • 8.2. Standard Model
    • 8.2.1. Electroweak Standard Model
    • 8.2.2. QFT
    • 8.2.3. Low energy theories
  • 8.3. Quantum Electrodynamics (QED)
    • 8.3.1. Local Gauge Invariance
    • 8.3.2. QED Lagrangian
    • 8.3.3. Magnetic moment of an electron
  • 8.4. Quantum Mechanics
    • 8.4.1. From QED to Quantum Mechanics
    • 8.4.2. Perturbation Theory
    • 8.4.3. Scattering Theory
  • 8.5. Systematic Perturbation Theory in QM
  • 8.6. Appendix
    • 8.6.1. Units and Dimensional Analysis
    • 8.6.2. Atomic Units
    • 8.6.3. Tensors in Special Relativity and QFT
    • 8.6.4. Adding Angular Momenta
  • 8.7. Examples
    • 8.7.1. Two Particles in Harmonic Potential
    • 8.7.2. Quantum Harmonic Oscillator
    • 8.7.3. Radial Schrödinger Equation
  • 8.8. Radial Schrödinger and Dirac Equations
    • 8.8.1. Variational Formulation of the Schrödinger equation
    • 8.8.2. Radial Schrödinger equation
    • 8.8.3. Variational Formulation of the Dirac equation
    • 8.8.4. Radial Dirac equation
    • 8.8.5. Other Forms of Dirac Equations
    • 8.8.6. Asymptotic
  • 8.9. Density Functional Theory (DFT)
    • 8.9.1. Many Body Schrödinger Equation
    • 8.9.2. The Hohenberg-Kohn Theorem
    • 8.9.3. The Kohn-Sham Equations
    • 8.9.4. The XC Term
    • 8.9.5. Total Energy
    • 8.9.6. XC Approximations
    • 8.9.7. Radial DFT Problem
    • 8.9.8. DFT As a Nonlinear Problem
    • 8.9.9. Thomas-Fermi-Dirac Theory
    • 8.9.10. Orbital Free Density Functional Theory
    • 8.9.11. References
  • 8.10. Ideal Fermi Gas
    • 8.10.1. Low Temperature Limit
    • 8.10.2. High Temperature Limit
  • 8.11. Hartree-Fock (HF) Method
    • 8.11.1. Derivation
    • 8.11.2. Roothaan Equations For Closed Shell Systems
    • 8.11.3. Two Particle Matrix Element
    • 8.11.4. General Matrix Elements in Spherical Symmetry
    • 8.11.5. Slater Type Orbitals (STO)
    • 8.11.6. Gaussian Type Orbitals (GTO)
    • 8.11.7. Exchange Integral in Spherical Symmetry
    • 8.11.8. Occupation Numbers
    • 8.11.9. Hartree Screening Functions
    • 8.11.10. Hartree Potential in Spherical Symmetry
    • 8.11.11. Nonlocal Exchange Potential in Spherical Symmetry
    • 8.11.12. Radial Hartree-Fock Equations
    • 8.11.13. Total Energy
    • 8.11.14. FEM
    • 8.11.15. 4-Index Transformation
    • 8.11.16. Green’s Functions
  • 8.12. Projector Augmented-Wave Method (PAW)
    • 8.12.1. Projectors, Augmentation Spheres and Smooth Wavefunctions
    • 8.12.2. Frozen Core Approximation
    • 8.12.3. Expectation Values of Local Operators
    • 8.12.4. Kohn Sham Equations