Quick Reference: Physics Formulas on the Cluster

A cheat sheet of key equations appearing across the wiki, organized by topic area.

Mechanics & Relativity

• Lagrangian: L = T - V (kinetic minus potential energy)
• Euler-Lagrange: d/dt (∂L/∂q_dot) - ∂L/∂q = 0
• Hamiltonian: H = p*q_dot - L
• Poisson bracket: {q, p} = 1
• Least action: δS = 0 where S = ∫ L dt
• Relativistic energy: E² = p²c² + m²c⁴
• Lorentz factor: γ = 1/√(1 - v²/c²)

Quantum Mechanics

• Schrödinger equation: iℏ ∂ψ/∂t = Ĥψ
• Uncertainty principle: ΔxΔp ≥ ℏ/2
• De Broglie wavelength: λ = h/p
• Born rule: P = |ψ|²
• Pauli exclusion: No two fermions in same quantum state
• Spin operators: σ_x, σ_y, σ_z (Pauli matrices)

Electromagnetism

• Maxwell's equations: ∇·E = ρ/ε₀, ∇×B = μ₀J + μ₀ε₀∂E/∂t, etc.
• Lorentz force: F = q(E + v × B)
• Gauss's law: Φ_E = Q_enclosed/ε₀
• Poynting vector: S = (1/μ₀) E × B
• Coulomb's law: F = k q₁q₂/r²

Thermodynamics & Statistical Mechanics

• First law: dU = dQ - dW
• Entropy: S = k_B ln Ω
• Boltzmann factor: P ∝ exp(-E/k_BT)
• Partition function: Z = Σ exp(-βE_i)
• Carnot efficiency: η = 1 - T_c/T_h
• Stefan-Boltzmann: j* = σT⁴

Particle Physics & QFT

• Fine structure constant: α ≈ 1/137
• Dirac equation: (iγ^μ∂_μ - m)ψ = 0
• Feynman propagator: D_F(x-y) = ⟨0|T{φ(x)φ(y)}|0⟩
• Gauge coupling running: μ dα/dμ = β(α)

Condensed Matter

• Band gap: E_gap = E_conduction - E_valence
• Fermi energy: E_F = ℏ²(3π²n)^(2/3)/(2m)
• Debye temperature: θ_D = ℏv_s(6π²n)^(1/3)/k_B
• Cooper pair binding: Δ ∝ exp(-1/N(0)V)

Astrophysics & Cosmology

• Hubble's law: v = H₀d
• Schwarzschild radius: R_s = 2GM/c²
• Hawking temperature: T_H = ℏc³/(8πGMk_B)
• Bekenstein-Hawking entropy: S = k_B A/(4l_P²)
• Chandrasekhar limit: M_ch ≈ 1.4 M_☉

About this reference: This page collects key formulas from across the wiki for quick lookup. Each formula appears in at least one linked page with deeper explanation.