Theoretical Physics Reference

• Introduction
  • Preface
  • Introduction
• Contributors
• Mathematics
  • Integration
    • General Case
    • Special Cases
    • Motivation
    • Example
  • Residue Theorem
    • Computation of Residues
    • Useful Formulas
    • Complex Substitution
  • Fourier Transform
  • Discrete Fourier Transform
  • Laplace Transform
  • Periodic Functions
  • Polar Coordinates
    • Example
  • Spherical Coordinates
    • Example I
    • Example II
  • Argument function, atan2
  • Multiple Argument Formulas
    • sin(a x)
    • cos(a x)
  • Delta Function
  • Distributions
  • Variations and Functional Derivatives
    • Functions of One Variable
    • Functions of several variables
    • Functionals
    • Examples
  • Dirac Notation
  • Homogeneous functions
    • Example 1
    • Example 2
  • Green Functions
    • Introduction
    • Boundary Conditions
    • Symmetry
    • Examples
  • Binomial Coefficients
  • Double Sums
  • Triangle Inequality
  • Gamma Function
  • Incomplete Gamma Function
    • Example
  • Factorial
  • Double Factorial
    • Example
  • Legendre Polynomials
    • Example I
    • Example II
    • Example III
    • Example IV
  • Spherical Harmonics
    • Examples
  • Gaunt Coefficients
    • Example I
    • Example II
    • Example III
    • Example IV
    • Example V
    • Example VI
  • Wigner 3j Symbols
    • Examples
  • Multipole Expansion
  • Hypergeometric Functions
    • Convergence Conditions
    • Elementary and Special Functions
    • Example I
    • Example II
    • Example III
  • Feynman Parameters
    • Example 1
    • Example 2
  • Groups
    • Theory
    • Crystallographic Point Groups
    • Applications of finite groups
    • Continuous Groups
    • Literature
  • Wigner D Function
    • Derivation
  • Ordinary Differential Equations
    • Finite Difference Formulas
    • Integrating ODE
    • Radial Poisson Equation
  • Linear Algebra
    • Scalar Product
    • Projections
  • Differential Geometry
    • Manifolds
    • Examples
  • Operators
    • Introduction
    • Spectrum
    • Derivative Operator
    • Sturm–Liouville Operator
    • Angular Momentum Operator
• Classical Mechanics, Special and General Relativity
  • Gravitation and Electromagnetism as a Field Theory
    • Gravitation
    • Electromagnetism
    • Relativistic Dust
    • Equations of Motion
  • Classical Mechanics
    • Rigid Body Rotation
  • Relativity
    • Introduction: Why Tensors
    • High School Formulation
    • College Formulation
    • Differential Geometry Formulation
    • Metrics
    • Conclusion About Metric
    • Einstein’s Equations
    • Continuous Distribution of Matter
    • Obsolete Section
    • Inertial frames
    • Lorentz Group
    • O(4) Group
    • Proper Time
    • FAQ
    • Questions Without Answers (Yet)
• Classical Electromagnetism
  • Maxwell’s Equations
    • Electromagnetic Field
    • Maxwell’s Equations
    • Lorentz Force
    • Hamiltonian
    • Electromagnetic Stress Tensor
    • Examples
  • Semiconductor Device Physics
    • Equations
    • Example 3
• Fluid Dynamics
  • Fluid Dynamics
    • Stress-Energy Tensor
    • Navier-Stokes Equations
    • Bernoulli’s Principle
  • MHD Equations
    • Introduction
    • Derivation
    • Finite Element Formulation
  • Compressible Euler Equations
    • Introduction
    • Dimensionless Euler Equations
    • Conservative Form of the Euler Equations
    • Weak Formulation
    • Flux Jacobians
    • 2D Version of the Equations
    • Sea Breeze Modeling
    • Boundary Conditions
    • Newton Method
    • Older notes
• Quantum Field Theory and Quantum Mechanics
  • Introduction
  • Standard Model
    • Electroweak Standard Model
    • QFT
    • Low energy theories
  • Quantum Electrodynamics (QED)
    • QED Lagrangian
    • Magnetic moment of an electron
  • Quantum Mechanics
    • From QED to Quantum Mechanics
    • Perturbation Theory
    • Scattering Theory
  • Systematic Perturbation Theory in QM
  • Appendix
    • Units and Dimensional Analysis
    • Atomic Units
    • Tensors in Special Relativity and QFT
    • Adding Angular Momenta
  • Examples
    • Two Particles in Harmonic Potential
    • Quantum Harmonic Oscillator
    • Radial Schrödinger Equation
  • Radial Schrödinger and Dirac Equations
    • Variational Formulation of the Schrödinger equation
    • Radial Schrödinger equation
    • Variational Formulation of the Dirac equation
    • Radial Dirac equation
    • Other Forms of Dirac Equations
    • Asymptotic
  • Density Functional Theory (DFT)
    • Many Body Schrödinger Equation
    • The Hohenberg-Kohn Theorem
    • The Kohn-Sham Equations
    • The XC Term
    • Total Energy
    • XC Approximations
    • Radial DFT Problem
    • DFT As a Nonlinear Problem
    • References
  • Hartree-Fock (HF) Method
    • Derivation
    • Roothaan Equations For Closed Shell Systems
    • Two Particle Matrix Element
    • General Matrix Elements in Spherical Symmetry
    • Slater Type Orbitals (STO)
    • Gaussian Type Orbitals (GTO)
    • Exchange Integral in Spherical Symmetry
    • Occupation Numbers
    • Hartree Screening Functions
    • Hartree Potential in Spherical Symmetry
    • Nonlocal Exchange Potential in Spherical Symmetry
    • Radial Hartree-Fock Equations
    • Total Energy
    • FEM
    • 4-Index Transformation
    • Green’s Functions
  • Projector Augmented-Wave Method (PAW)
    • Projectors, Augmentation Spheres and Smooth Wavefunctions
    • Frozen Core Approximation
    • Expectation Values of Local Operators
    • Kohn Sham Equations