GRE Physics Subject Test cheat sheet

Newton's second law: F = dp/dt; for constant mass F = ma. In non-inertial frames add fictitious force −ma_frame.

Energy: T = ½mv², U_grav = mgh near Earth, U_spring = ½kx². Total mechanical energy conserved only when all forces are conservative.

Momentum: p = mv; impulse J = ∫F dt = Δp. Total momentum conserved when net external force is zero, including collisions.

Angular momentum: L = r × p = Iω; torque τ = dL/dt. For a rigid body about a fixed axis, τ = Iα.

Moment of inertia: I = ∫r² dm; parallel-axis theorem I = I_cm + Md². Solid sphere 2MR²/5, hoop MR², rod about center ML²/12.

Simple harmonic motion: x'' + ω²x = 0 with ω = √(k/m) for a spring and ω = √(g/L) for a small-angle pendulum; period T = 2π/ω.

Orbits and gravitation: F = GmM/r²; circular orbit v = √(GM/r); Kepler's third law T² ∝ a³; escape speed √(2GM/R).

Central-force motion: effective potential U_eff = U(r) + L²/(2mr²); circular orbits occur at minima of U_eff.

Lagrangian mechanics: L = T − U; Euler–Lagrange equation d/dt(∂L/∂q̇) − ∂L/∂q = 0; cyclic coordinates give conserved conjugate momenta.

Coulomb's law: F = kq₁q₂/r² with k = 1/(4πε₀) ≈ 9×10⁹ N·m²/C²; field of a point charge E = kq/r².

Gauss's law: ∮E·dA = Q_enc/ε₀. Symmetric charge distributions (sphere, cylinder, sheet) give E from enclosed charge alone.

Potential: V = kq/r for a point charge; E = −∇V. Work done by the field on a charge q moving through ΔV is qΔV (with sign).

Capacitance: C = Q/V; parallel plate C = ε₀A/d; energy stored U = ½CV² = Q²/(2C); dielectrics multiply C by κ.

Circuits: resistors in series add; in parallel conductances add. Kirchhoff loop rule ΣΔV = 0 and junction rule ΣI = 0. RC time constant τ = RC.

Magnetic force: F = qv × B; a wire carries force F = IL × B. Moving charges create B = μ₀I/(2πr) for a long wire.

Ampère's law: ∮B·dl = μ₀I_enc (+ μ₀ε₀ dΦ_E/dt for the displacement current). Solenoid interior field B = μ₀nI.

Faraday's law: emf = −dΦ_B/dt; motional emf = Bℓv for a rod moving perpendicular to B. Lenz's law fixes the direction opposing flux change.

Maxwell's equations in vacuum give electromagnetic waves with speed c = 1/√(μ₀ε₀); E and B are perpendicular, in phase, with E = cB.

Wave equation: ∂²y/∂t² = v²∂²y/∂x²; solutions y = A sin(kx − ωt) with v = ω/k = fλ.

Superposition: for two coherent sources, constructive interference requires path difference mλ, destructive requires (m + ½)λ.

Double-slit pattern: bright fringes at d sinθ = mλ; fringe spacing on a distant screen Δy ≈ λL/d.

Single-slit diffraction: minima at a sinθ = mλ (m ≠ 0); first minimum half-width λL/a. Rayleigh criterion θ ≈ 1.22λ/D for a circular aperture.

Thin-film interference: phase shift of π occurs on reflection from a higher-index medium; net condition depends on the number of such shifts.

Diffraction grating: maxima at d sinθ = mλ; resolving power R = mN.

Snell's law: n₁ sinθ₁ = n₂ sinθ₂; total internal reflection when sinθ_c = n₂/n₁ with n₁ > n₂.

Lenses and mirrors: 1/f = 1/d_o + 1/d_i; magnification m = −d_i/d_o. Converging lens f > 0; diverging lens f < 0.

Polarization: Malus's law I = I₀cos²θ through a polarizer; Brewster angle satisfies tanθ_B = n₂/n₁.

First law: ΔU = Q − W, with W = ∫P dV the work done by the gas. For an ideal gas U depends only on T.

Ideal gas law: PV = nRT = Nk_BT; average kinetic energy per molecule (3/2)k_BT for a monatomic gas.

Specific heats: C_P − C_V = nR for an ideal gas; γ = C_P/C_V equals 5/3 monatomic and 7/5 diatomic.

Isothermal work W = nRT ln(V₂/V₁); adiabatic relation PV^γ = constant; efficiency of a Carnot cycle η = 1 − T_c/T_h.

Second law: entropy of an isolated system never decreases; ΔS = ∫dQ_rev/T. Reversible adiabatic processes are isentropic.

Entropy of an ideal gas: ΔS = nC_V ln(T₂/T₁) + nR ln(V₂/V₁). Free expansion of an ideal gas has ΔS > 0 with ΔU = 0.

Boltzmann distribution: probability ∝ exp(−E/k_BT); equipartition gives ½k_BT per quadratic degree of freedom.

Partition function Z = Σ exp(−E_i/k_BT); F = −k_BT ln Z; mean energy from −∂lnZ/∂β with β = 1/k_BT.

Maxwell–Boltzmann speed distribution peaks at √(2k_BT/m); mean speed √(8k_BT/πm); rms speed √(3k_BT/m).

de Broglie relations: p = h/λ and E = hf; ħ = h/2π. Photon energy E = hc/λ ≈ 1240 eV·nm/λ(nm).

Uncertainty principle: ΔxΔp ≥ ħ/2 and ΔEΔt ≥ ħ/2.

Schrödinger equation: iħ∂ψ/∂t = Ĥψ; time-independent form Ĥψ = Eψ. |ψ|² is a probability density.

Infinite square well of width L: E_n = n²π²ħ²/(2mL²); wavefunctions are sine functions vanishing at the walls.

Harmonic oscillator: E_n = (n + ½)ħω; ground state has zero-point energy ½ħω.

Hydrogen atom: E_n = −13.6 eV/n²; angular momentum magnitude √(l(l+1))ħ; m_l takes 2l + 1 values.

Tunneling: transmission through a rectangular barrier depends exponentially on barrier width and √(2m(V − E))/ħ.

Operators: momentum p̂ = −iħ∇; expectation values ⟨A⟩ = ∫ψ*Aψ dV. Commuting observables share eigenstates; [x, p] = iħ.

Spin: S_z eigenvalues ±ħ/2 for a spin-½ particle; identical fermions obey the Pauli exclusion principle and antisymmetric wavefunctions.

Bohr model: E_n = −13.6 eV/n² for hydrogen; transition photon energy equals the level difference.

Rydberg formula: 1/λ = R(1/n_f² − 1/n_i²) with R ≈ 1.097×10⁷ m⁻¹.

Lyman series ends on n = 1 (ultraviolet); Balmer series ends on n = 2 (visible); Paschen ends on n = 3 (infrared).

Quantum numbers: n ≥ 1, 0 ≤ l ≤ n − 1, −l ≤ m_l ≤ l, m_s = ±½; shell capacity 2n².

Fine structure arises from spin–orbit coupling and relativistic corrections; it splits levels by total angular momentum j.

Selection rules for electric dipole transitions: Δl = ±1, Δm_l = 0, ±1; Δn unrestricted.

X-ray spectra: Moseley's law √f ∝ (Z − 1) for Kα lines; characteristic lines come from inner-shell vacancies.

Photoelectric effect: maximum kinetic energy K_max = hf − φ, with threshold frequency f₀ = φ/h.

Multi-electron atoms: Hund's rules give ground-state term ordering; ionization energy generally rises across a period but drops at noble-gas-to-alkali boundaries.

Lorentz factor γ = 1/√(1 − v²/c²); γ ≥ 1 and grows without bound as v → c.

Time dilation: Δt = γΔτ for a clock moving relative to the observer; the proper time is measured in the clock's rest frame.

Length contraction: L = L₀/γ along the direction of motion; transverse dimensions are unchanged.

Lorentz transformations: x' = γ(x − vt), t' = γ(t − vx/c²); velocities add as u' = (u − v)/(1 − uv/c²).

Relativistic momentum p = γmv; total energy E = γmc²; rest energy E₀ = mc²; E² = p²c² + m²c⁴.

Kinetic energy K = (γ − 1)mc²; for v ≪ c this reduces to ½mv².

Invariant interval: s² = c²t² − x² − y² − z² is the same in all inertial frames; spacelike, timelike, and lightlike separations are frame-independent.

Relativistic Doppler effect: f_obs = f_source√((1 − β)/(1 + β)) for a receding source, with β = v/c.

Massless particles travel at c and satisfy E = pc; they cannot be brought to rest in any frame.

Uncertainty propagation: for independent random errors, σ_f² = Σ(∂f/∂x_i)²σ_i². Add relative errors in quadrature for products and quotients.

Standard error of the mean is σ/√N; increasing sample count reduces random, not systematic, uncertainty.

Least-squares fit of y = mx + b: slope m = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)²; the intercept and slope uncertainties come from residuals.

Error bars and fits: a parameter is consistent with a model if the difference is within roughly two combined standard deviations.

Resolution and accuracy: instrument resolution sets the smallest distinguishable change; calibration removes systematic offsets.

Counting statistics: radioactive counts follow a Poisson distribution with σ = √N; fractional uncertainty falls as 1/√N.

Oscilloscope and multimeter use: voltage measured in parallel, current in series; a high-impedance voltmeter minimizes circuit loading.

Spectroscopy: a diffraction grating or prism disperses wavelengths; calibration with known spectral lines identifies unknown lines.

Interferometry: fringe shifts of one full fringe correspond to a path change of one wavelength; Michelson setups measure small displacements and index changes.

Nuclear binding: mass defect Δm gives binding energy Δmc²; the curve peaks near A ≈ 56, so fusion releases energy for light nuclei and fission for heavy ones.

Radioactive decay: N = N₀e^(−λt); half-life t½ = ln2/λ; activity A = λN.

Alpha, beta, and gamma decay differ in charge and penetration; beta decay conserves lepton number and emits a neutrino.

Particle physics: quarks carry fractional charge (+2/3 for u, c, t; −1/3 for d, s, b); hadrons are color-singlet bound states.

Conservation laws in reactions: energy–momentum, electric charge, baryon number, and lepton number; strangeness is conserved in strong interactions only.

Solid-state basics: Fermi–Dirac statistics; Fermi energy at T = 0 fills states up to E_F; semiconductors have a band gap of order 1 eV.

Superconductivity: zero resistance below T_c and expulsion of magnetic flux (Meissner effect); BCS pairs electrons via lattice vibrations.

General relativity: equivalence principle; gravitational redshift and time dilation near a mass; Schwarzschild radius r_s = 2GM/c².

Cosmology: Hubble's law v = H₀d; cosmic microwave background temperature ≈ 2.7 K; redshift z relates to scale factor as 1 + z = a₀/a.