As for myself, like the character in Molière’s play Le Bourgeois Gentilhomme who suddenly discovers that he has been speaking in prose all these years without knowing it, I realized that I had been doing theoretical physics without knowing it.

These are miscellaneous theoretical derivations and calculations.

Physics

Mechanics

• Damped Simple Harmonic Oscillator

• Hamilton’s Equations of Motion

• Escape Velocity

• Newtonian Gravitational Deflection of Light

• Scattering Cross Section of Hard Sphere Target

• 1D Elastic Collision with Initially Stationary Particle

• Virial Theorem

• Relevance of the Simple Harmonic Oscillator Model

• Evolution of Two Point Particles Under Coulomb Force in 1D

• Total Linear Momentum Vanishes in the Center of Mass Frame

Statistical Mechanics, Thermodynamics, and Kinetic Theory

• Heat Capacity of the Ideal Gas

• Heat Capacity of the Einstein Solid (a.k.a. the Dulong-Petit law)

• Phase Transition in the Landau Model of Ferromagnetism

• No Phase Transitions in the 1D Ising Model

• Bose-Einstein Distribution

• High Temperature Equation of State of a Bose Gas

• Density of States for Quantum Particle in 3D Box with Periodic BCs

• Density of States for Relativistic Quantum Particle in 3D Box with Periodic BCs (Massive and Massless)

• Heat Capacity of Solids in the Debye Model

• Heat Capacity of Diatomic Gas

• Heat Capacity of Diatomic Gas vs. Temperature

• Thermal Expansion of Materials

• Equipartition Theorem

• Two-State System Thermodynamics

• Nearest Neighbor Distribution in n-Dimensions

• Shannon Entropy in the Canonical Ensemble

• Laminar Flow in a Cylindrical Pipe for Various Viscosities

• Free-Space Diffusion Equation in 1D with Delta-Function Initial Condition

• Thermal de Broglie Wavelength from the Equipartition Theorem

• When Must The Ideal Gas Be Treated Quantum Mechanically?

• Diffusion in a Very Viscous Fluid

• Hard Sphere Particle-Particle Collision Time

• Hard Sphere Particle-Wall Collision Time

Quantum Mechanics

• [a_,a+]=1 –– Commutator of raising and lowering operators

• Wave Function of Quantum Particle in 3D Box with Periodic BCs

• Slater Determinant –– Single-Particle Wavefunctions are Linearly Independent

• Slater Determinant –– Normalization

• Slater Determinant –– Expectation Value of Arbitrary Single-Particle Potential

• Slater Determinant –– Expectation Value of Arbitrary Pairwise Potential

• Functional Derivative of the Expectation Value of the Hartree-Fock Hamiltonian in the Slater Determinant State

• Hartree-Fock Equation

• Radial Schrödinger Equation

• 3D Quantum Harmonic Oscillator Spectrum

• Larmor Precession

• Quantum Harmonic Oscillator Algebra

• 1D Finite Square Well Bound States

• Change of Basis in QM (Unitary Transformations)

• Eigenvectors of Hermitian Operators Are Orthonormal

• Variational Principle

• First-Order Nondegenerate Perturbation Theory –– Energy Correction

• Translation Operator

• Probability Conservation in Quantum Mechanics

• Momentum Operator in Quantum Mechanics

• Ehrenfest’s Theorem

• Conservation of Probability Current in Quantum Mechanics

• Wave Function Overlaps are Preserved by Schrödinger Evolution

• Unstable Particle in Quantum Mechanics

• Time-Independent Schrodinger Equation

• Infinite Square Well in 1D

• Infinite Square Well Eigenstates are Orthonormal

• Fourier’s Trick

• Quantum Mechanics Violates Causality

• Path Integral in Quantum Mechanics

• Action of Raising and Lowering Operators on Harmonic Oscillator Eigenstates

Quantum Field Theory

• Principle of Least Action for Fields

• Klein-Gordon Equation

• Solutions to the Dirac Equation Satisfy the Klein-Gordon Equation

Electricity and Magnetism

• Classical Atoms Are Unstable

• Electric Field and Potential of Uniformly Charged Sphere

• Electric Field and Potential of Exponentially Decaying Charge Distribution

• Electrostatic Potential Energy

• Multipole Expansion (Electrostatics)

Relativity

• Relativistic Energy

• Time Dilation and Length Contraction

• 1D Relativistic Velocity Addition

• Circular Orbits in the Schwarzchild Metric

Miscellaneous

• Snell’s Law

• Green’s Function for Simple Harmonic Oscillator

• Compton Scattering

• The Wave Equation

• Green’s Function for Inhomogeneous Differential Equations

  
  
  

Math & Statistics

• Quadratic Formula

• Series –– 1/n

• Series –– 1 - 1/2 + 1/3 - 1/4 + … = (-1)^n/(n+1)

• det(A) = det(A^T)

• Geometric Series Formula –– 1 + x + x^2 + x^3 + …

• Finite Geometric Series Formula Starting from Negative Index

• Product Rule

• Eigenvalues of Hermitian Matrices Are Real

• Fourier Transform of (Offset) Gaussian is a Gaussian

• Change of Basis Formula for Tangent Vectors in Differential Geometry

• Spherical Unit Vectors

• Gradient in Spherical Coordinates

• The Gamma Function

• Linear Regression

• Newton’s Method

• Orthogonality of Sine and Cosine

• Cramer’s Rule

• Odd Moments of Symmetric Probability Distributions Vanish

• Mean = Median for 1D Symmetric Probability Distributions

• Mean and Variance of Binomial Distribution

• Lagrange Multipliers for Maximum Entropy Inference

• LU Decomposition

• Shannon Entropy

• Differentiating Integral with Function in Bound

• Linear Regression with Weights

• MLE for Power Law Distribution

• MLE for Maxwell-Boltzmann Distribution

• Uncertainty in MLE for Distribution Parameter

• Gaps Between Primes Can Be Arbitrarily Large

Algorithms and Code

• Bisection Method (Python)

• Secant Method (Python)

• Simulated Annealing (Python)

• Hard Disk Monte Carlo Simulation (C)

• Compute all pairwise distances in 2D with periodic boundary conditions (C)

• Hard disk forward-Euler simulation with periodic boundary conditions (C)

• Ising model (Python)

Problems I Particularly Like

(Very rough at present.)

• Solving the Poisson equation for a point particle

• Solve the time-dependent Schrodinger equation for a delta-function initial state

• Electrostatic energy of a point charge