# 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_☉

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**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.