Physics GRE Equation Sheet

Physics GRE equation sheet

This Physics GRE equation sheet (also a formula sheet and study guide) is a formula-first review for the PGRE: classical mechanics, electromagnetism, quantum mechanics, thermodynamics/statistical mechanics, optics, relativity, laboratory methods, and specialized topics. Keep it open while you drill timed sets.

Approximate topic distribution

Classical mechanics formula sheet

Newton II:                 F = dp/dt = ma
Work–energy:               W = ∫ F·dr = ΔK
Conservation:              E = K + U  (conservative forces)
Simple harmonic motion:    x(t) = A cos(ωt + φ),  ω = √(k/m)
Physical pendulum:         ω = √(mgd/I)
Central force orbits:      L = μ r² θ̇  conserved
Kepler III:                T² = (4π²/GM) a³
Lagrangian:                L = T − V,  d/dt(∂L/∂q̇) = ∂L/∂q
Hamiltonian:               H = p q̇ − L

Electromagnetism essentials

Coulomb / field:           E = (1/4πε₀) q r̂ / r²
Gauss’s law:               ∮ E·dA = Q_enc / ε₀
Potential:                 V = −∫ E·dl ,  E = −∇V
Biot–Savart:               dB = (μ₀/4π) I dl × r̂ / r²
Ampère–Maxwell:            ∮ B·dl = μ₀(I_enc + ε₀ dΦ_E/dt)
Faraday:                   ∮ E·dl = −dΦ_B/dt
Poynting:                  S = (1/μ₀) E × B
Wave speed:                c = 1/√(μ₀ε₀)
AC impedance:              Z = R + i(ωL − 1/ωC)

For a deeper conceptual walkthrough of electrostatic fields, see the Electric Field Guide & Visualizer.

Quantum mechanics checklist

• Postulates: states as kets, observables as Hermitian operators, Born rule probabilities.
• Infinite well: E_n = n²π²ℏ² / (2mL²), nodes = n−1.
• Harmonic oscillator: E_n = ℏω(n + 1/2).
• Hydrogen: E_n = −13.6 eV / n², degeneracy n² (ignoring spin).
• Commutators: [x, p] = iℏ, angular momentum algebra [J_i, J_j] = iℏ ε_ijk J_k.
• Spin-1/2: Pauli matrices, Stern–Gerlach intuition, addition of angular momenta.

Thermodynamics & statistical mechanics

First law:                 ΔU = Q − W  (sign convention dependent)
Ideal gas:                 PV = NkT = nRT
Entropy (Clausius):        dS = đQ_rev / T
Maxwell–Boltzmann:         f(v) ∝ v² exp(−mv²/2kT)
Partition function:        Z = Σ_i e^{−βE_i},  β = 1/kT
Helmholtz free energy:     F = −kT ln Z = U − TS
Equipartition:             (1/2)kT per quadratic degree of freedom

Optics & waves formula sheet

Wave:                      v = fλ
Intensity (wave):          I ∝ A²
Snell:                     n₁ sin θ₁ = n₂ sin θ₂
Thin lens:                 1/f = 1/s + 1/s′
Magnification:             m = −s′/s = h′/h
Double slit:               d sin θ = mλ  (bright)
Single-slit min:           a sin θ = mλ
Diffraction grating:       d sin θ = mλ
Rayleigh criterion:        θ ≈ 1.22 λ/D
Doppler (sound, source):   f′ = f v/(v ± v_s)

Special relativity

Lorentz factor:            γ = 1/√(1 − β²),  β = v/c
Time dilation:             Δt = γ Δτ
Length contraction:        L = L₀/γ
Velocity addition:         u = (v + u′)/(1 + vu′/c²)
Energy–momentum:           E² = (pc)² + (mc²)²
Rest energy:               E₀ = mc²
Kinetic energy:            K = (γ − 1)mc²

High-yield constants

c = 3.00×10⁸ m/s
h = 6.63×10⁻³⁴ J·s    ℏc ≈ 197 MeV·fm
k = 8.62×10⁻⁵ eV/K    N_A = 6.02×10²³ mol⁻¹
e = 1.60×10⁻¹⁹ C      α ≈ 1/137
m_e = 511 keV/c²      m_p ≈ 938 MeV/c²
g ≈ 9.8 m/s²          σ = 5.67×10⁻⁸ W·m⁻²·K⁻⁴
ε₀ = 8.85×10⁻¹² F/m   μ₀ = 4π×10⁻⁷ T·m/A

High-yield exam strategy

1. Drill dimensional analysis and limiting cases first — many PGRE items reward quick elimination.
2. Memorize order-of-magnitude constants (ℏc ≈ 197 MeV·fm, k ≈ 8.6×10⁻⁵ eV/K, α ≈ 1/137).
3. Practice under timed conditions — roughly 1.7 minutes per question across ~100 items.
4. Rotate weak topics weekly instead of re-reading only mechanics comfort zones.
5. Use interactive tools while reviewing — plot potentials with the graphing calculator and explore ODE phase structure with the phase portrait generator.

Two-week intensive outline

• Days 1–3: Mechanics + oscillations + central forces
• Days 4–6: E&M (electrostatics through Maxwell)
• Days 7–8: Quantum + atomic spectra
• Days 9–10: Thermo/stat mech + lab methods
• Days 11–12: Optics, waves, relativity
• Days 13–14: Full practice exams + error logs

For hands-on practice visualization, use the graphing calculator to plot potentials and waveforms, the phase portrait generator for dynamical systems, and the time graphing tool for position–velocity–time curves.

Bookmark this equation sheet, then jump into practice sets and interactive visualizations to convert recognition into speed.