APS Graduate Exam cheat sheet

Dimensional analysis: every physical equation must have identical dimensions on both sides; use [M][L][T] exponents to check or derive relations.

SI base units: meter, kilogram, second, ampere, kelvin, mole, candela; derived units are products of these (1 N = 1 kg·m·s^-2, 1 J = 1 N·m).

Waves: v = fλ; for a wave on a string v = sqrt(T/μ), for light in a medium v = c/n.

Superposition and interference: path difference Δr = mλ gives constructive, Δr = (m+1/2)λ gives destructive interference.

Doppler shift for sound: f' = f (v ± v_o)/(v ∓ v_s), with signs set by motion toward or away from the observer.

Snell's law: n1 sin θ1 = n2 sin θ2; total internal reflection occurs when θ1 exceeds θ_c = arcsin(n2/n1) for n1 > n2.

Photoelectric effect: K_max = hf − φ, with threshold frequency f_0 = φ/h; no emission below f_0 regardless of intensity.

Uncertainty propagation: for independent random errors, fractional errors add in quadrature; for z = x^a y^b, (σ_z/z)^2 = a^2(σ_x/x)^2 + b^2(σ_y/y)^2.

Newton's second law in inertial frames: F_net = dp/dt = ma for constant mass; in non-inertial frames add fictitious forces (−ma_frame).

Work–energy theorem: W_net = ΔK, with K = ½mv²; for conservative forces define U by F = −∇U and total E = K + U is conserved.

Momentum conservation holds when net external force is zero; collisions conserve momentum always, kinetic energy only if elastic.

Rotational dynamics: τ = Iα, L = Iω; for a rigid body about a fixed axis, I = Σ m_i r_i², and parallel-axis theorem I = I_cm + Md².

Angular momentum is conserved when net external torque vanishes; a spinning skater's ω scales as 1/I.

Simple harmonic motion: ẍ = −ω²x gives x = A cos(ωt + φ) with ω = sqrt(k/m) for a spring and ω = sqrt(g/L) for a small-angle pendulum.

Central-force orbits: effective potential U_eff = U(r) + L²/(2mr²); circular orbit requires dU_eff/dr = 0.

Kepler's third law for a two-body system: T² = 4π²a³/(G(M+m)); for planets about the Sun, T² ∝ a³.

Lagrangian mechanics: L = T − U, and the Euler–Lagrange equations d/dt(∂L/∂q̇) − ∂L/∂q = 0 generate the equations of motion.

Coulomb's law: F = k q1 q2 / r² with k = 1/(4πε0) ≈ 8.99×10^9 N·m²·C^-2; force is along the line joining charges.

Gauss's law: ∮E·dA = Q_enc/ε0; use symmetry (spherical, cylindrical, planar) to extract E for highly symmetric charge distributions.

Electric potential: V = −∫E·dl and E = −∇V; equipotential surfaces are perpendicular to field lines.

Capacitance: C = Q/V; parallel plate C = ε0 A/d, energy stored U = ½CV² = Q²/(2C); dielectrics multiply C by κ.

Lorentz force: F = q(E + v×B); magnetic forces do no work, so they change direction but not speed of a charged particle.

Biot–Savart and Ampère: ∮B·dl = μ0 I_enc; a long straight wire gives B = μ0 I/(2πr).

Faraday's law: EMF = −dΦ_B/dt; Lenz's law fixes the induced current direction to oppose the change in flux.

Maxwell's equations in vacuum predict EM waves with speed c = 1/sqrt(μ0 ε0); E, B, and k are mutually perpendicular with E = cB.

RC circuits: charging q(t) = Q(1 − e^(−t/RC)) with time constant τ = RC; RL circuits have τ = L/R.

de Broglie relation: λ = h/p; photon energy E = hf = hc/λ with hc ≈ 1240 eV·nm.

Heisenberg uncertainty: Δx Δp ≥ ħ/2 and ΔE Δt ≥ ħ/2; these are statistical spreads, not measurement disturbances.

Schrödinger equation: iħ ∂ψ/∂t = Ĥψ; stationary states satisfy Ĥψ = Eψ with |ψ|² the probability density.

Infinite square well of width L: E_n = n²π²ħ²/(2mL²), ψ_n = sqrt(2/L) sin(nπx/L); nodes increase with n.

Harmonic oscillator: E_n = (n + ½)ħω; zero-point energy ½ħω is nonzero, unlike the classical minimum.

Hydrogen atom: E_n = −13.6 eV/n²; orbital angular momentum magnitude L = sqrt(l(l+1))ħ with l < n.

Commutation and measurement: [x, p] = iħ; observables commute iff they can be simultaneously diagonalized and measured to arbitrary precision.

Identical particles: fermions obey antisymmetric wavefunctions and the Pauli exclusion principle; bosons obey symmetric wavefunctions.

Tunneling: a particle penetrates a barrier of height V0 > E with transmission decaying roughly as exp(−2κa), κ = sqrt(2m(V0−E))/ħ.

First law: ΔU = Q − W, with W = ∫P dV the work done by the system; U is a state function, Q and W are not.

Ideal gas law: PV = nRT = Nk_B T; average translational kinetic energy per molecule = (3/2)k_B T.

Entropy: dS = δQ_rev/T; the second law states ΔS_total ≥ 0 for any process, with equality only for reversible ones.

Carnot efficiency: η = 1 − T_c/T_h for a reversible engine operating between those reservoirs; no engine exceeds it.

Maxwell–Boltzmann distribution: number density ∝ exp(−E/k_B T); the equipartition theorem assigns ½k_B T per quadratic degree of freedom.

Statistical definitions: S = k_B ln Ω (microcanonical); partition function Z = Σ exp(−E_i/k_B T) with F = −k_B T ln Z.

Fermi–Dirac and Bose–Einstein distributions reduce to Maxwell–Boltzmann in the dilute limit; chemical potential controls occupation.

Maxwell relations follow from equality of mixed partials of thermodynamic potentials (U, H, F, G); e.g., (∂T/∂V)_S = −(∂P/∂S)_V.

Phase equilibrium requires equal temperature, pressure, and chemical potential; the Clausius–Clapeyron relation gives dP/dT = L/(TΔv).

Linear algebra: a matrix is diagonalizable iff it has a complete set of eigenvectors; Hermitian matrices have real eigenvalues and orthogonal eigenvectors.

Taylor expansion: f(x) = f(a) + f'(a)(x−a) + ½f''(a)(x−a)² + …; small-angle approximations sin θ ≈ θ and cos θ ≈ 1 − θ²/2 follow from it.

Fourier transform pair: f(t) = (1/2π)∫F(ω)e^(iωt)dω and F(ω) = ∫f(t)e^(−iωt)dt; narrow in time implies broad in frequency.

Vector calculus identities: ∇·(∇×A) = 0 and ∇×(∇f) = 0; the divergence and Stokes theorems convert between volume/surface and surface/line integrals.

Ordinary differential equations: a linear second-order ODE with constant coefficients is solved by the characteristic equation; damping regimes depend on the discriminant.

Error analysis: standard error of the mean is σ/√N; combining independent measurements uses quadrature, and a 1/√N scaling sets the precision floor.

Significant figures and uncertainty reporting: quote the uncertainty to one or two significant digits and match the value's last digit to it.

Experimental design: randomize and control variables, calibrate instruments against known standards, and distinguish systematic bias from random scatter.

Least-squares fitting: slope uncertainty scales inversely with the spread of x; χ² per degree of freedom near 1 indicates a consistent model and error estimate.

Dimensional analysis: every physical equation must have identical dimensions on both sides; use [M][L][T] exponents to check or derive relations.

SI base units: meter, kilogram, second, ampere, kelvin, mole, candela; derived units are products of these (1 N = 1 kg·m·s^-2, 1 J = 1 N·m).

Waves: v = fλ; for a wave on a string v = sqrt(T/μ), for light in a medium v = c/n.

Superposition and interference: path difference Δr = mλ gives constructive, Δr = (m+1/2)λ gives destructive interference.

Doppler shift for sound: f' = f (v ± v_o)/(v ∓ v_s), with signs set by motion toward or away from the observer.

Snell's law: n1 sin θ1 = n2 sin θ2; total internal reflection occurs when θ1 exceeds θ_c = arcsin(n2/n1) for n1 > n2.

Photoelectric effect: K_max = hf − φ, with threshold frequency f_0 = φ/h; no emission below f_0 regardless of intensity.

Uncertainty propagation: for independent random errors, fractional errors add in quadrature; for z = x^a y^b, (σ_z/z)^2 = a^2(σ_x/x)^2 + b^2(σ_y/y)^2.

Newton's second law in inertial frames: F_net = dp/dt = ma for constant mass; in non-inertial frames add fictitious forces (−ma_frame).

Work–energy theorem: W_net = ΔK, with K = ½mv²; for conservative forces define U by F = −∇U and total E = K + U is conserved.

Momentum conservation holds when net external force is zero; collisions conserve momentum always, kinetic energy only if elastic.

Rotational dynamics: τ = Iα, L = Iω; for a rigid body about a fixed axis, I = Σ m_i r_i², and parallel-axis theorem I = I_cm + Md².

Angular momentum is conserved when net external torque vanishes; a spinning skater's ω scales as 1/I.

Simple harmonic motion: ẍ = −ω²x gives x = A cos(ωt + φ) with ω = sqrt(k/m) for a spring and ω = sqrt(g/L) for a small-angle pendulum.

Central-force orbits: effective potential U_eff = U(r) + L²/(2mr²); circular orbit requires dU_eff/dr = 0.

Kepler's third law for a two-body system: T² = 4π²a³/(G(M+m)); for planets about the Sun, T² ∝ a³.

Lagrangian mechanics: L = T − U, and the Euler–Lagrange equations d/dt(∂L/∂q̇) − ∂L/∂q = 0 generate the equations of motion.

Coulomb's law: F = k q1 q2 / r² with k = 1/(4πε0) ≈ 8.99×10^9 N·m²·C^-2; force is along the line joining charges.

Gauss's law: ∮E·dA = Q_enc/ε0; use symmetry (spherical, cylindrical, planar) to extract E for highly symmetric charge distributions.

Electric potential: V = −∫E·dl and E = −∇V; equipotential surfaces are perpendicular to field lines.

Capacitance: C = Q/V; parallel plate C = ε0 A/d, energy stored U = ½CV² = Q²/(2C); dielectrics multiply C by κ.

Lorentz force: F = q(E + v×B); magnetic forces do no work, so they change direction but not speed of a charged particle.

Biot–Savart and Ampère: ∮B·dl = μ0 I_enc; a long straight wire gives B = μ0 I/(2πr).

Faraday's law: EMF = −dΦ_B/dt; Lenz's law fixes the induced current direction to oppose the change in flux.

Maxwell's equations in vacuum predict EM waves with speed c = 1/sqrt(μ0 ε0); E, B, and k are mutually perpendicular with E = cB.

RC circuits: charging q(t) = Q(1 − e^(−t/RC)) with time constant τ = RC; RL circuits have τ = L/R.

de Broglie relation: λ = h/p; photon energy E = hf = hc/λ with hc ≈ 1240 eV·nm.

Heisenberg uncertainty: Δx Δp ≥ ħ/2 and ΔE Δt ≥ ħ/2; these are statistical spreads, not measurement disturbances.

Schrödinger equation: iħ ∂ψ/∂t = Ĥψ; stationary states satisfy Ĥψ = Eψ with |ψ|² the probability density.

Infinite square well of width L: E_n = n²π²ħ²/(2mL²), ψ_n = sqrt(2/L) sin(nπx/L); nodes increase with n.

Harmonic oscillator: E_n = (n + ½)ħω; zero-point energy ½ħω is nonzero, unlike the classical minimum.

Hydrogen atom: E_n = −13.6 eV/n²; orbital angular momentum magnitude L = sqrt(l(l+1))ħ with l < n.

Commutation and measurement: [x, p] = iħ; observables commute iff they can be simultaneously diagonalized and measured to arbitrary precision.

Identical particles: fermions obey antisymmetric wavefunctions and the Pauli exclusion principle; bosons obey symmetric wavefunctions.

Tunneling: a particle penetrates a barrier of height V0 > E with transmission decaying roughly as exp(−2κa), κ = sqrt(2m(V0−E))/ħ.

First law: ΔU = Q − W, with W = ∫P dV the work done by the system; U is a state function, Q and W are not.

Ideal gas law: PV = nRT = Nk_B T; average translational kinetic energy per molecule = (3/2)k_B T.

Entropy: dS = δQ_rev/T; the second law states ΔS_total ≥ 0 for any process, with equality only for reversible ones.

Carnot efficiency: η = 1 − T_c/T_h for a reversible engine operating between those reservoirs; no engine exceeds it.

Maxwell–Boltzmann distribution: number density ∝ exp(−E/k_B T); the equipartition theorem assigns ½k_B T per quadratic degree of freedom.

Statistical definitions: S = k_B ln Ω (microcanonical); partition function Z = Σ exp(−E_i/k_B T) with F = −k_B T ln Z.

Fermi–Dirac and Bose–Einstein distributions reduce to Maxwell–Boltzmann in the dilute limit; chemical potential controls occupation.

Maxwell relations follow from equality of mixed partials of thermodynamic potentials (U, H, F, G); e.g., (∂T/∂V)_S = −(∂P/∂S)_V.

Phase equilibrium requires equal temperature, pressure, and chemical potential; the Clausius–Clapeyron relation gives dP/dT = L/(TΔv).

Linear algebra: a matrix is diagonalizable iff it has a complete set of eigenvectors; Hermitian matrices have real eigenvalues and orthogonal eigenvectors.

Taylor expansion: f(x) = f(a) + f'(a)(x−a) + ½f''(a)(x−a)² + …; small-angle approximations sin θ ≈ θ and cos θ ≈ 1 − θ²/2 follow from it.

Fourier transform pair: f(t) = (1/2π)∫F(ω)e^(iωt)dω and F(ω) = ∫f(t)e^(−iωt)dt; narrow in time implies broad in frequency.

Vector calculus identities: ∇·(∇×A) = 0 and ∇×(∇f) = 0; the divergence and Stokes theorems convert between volume/surface and surface/line integrals.

Ordinary differential equations: a linear second-order ODE with constant coefficients is solved by the characteristic equation; damping regimes depend on the discriminant.

Error analysis: standard error of the mean is σ/√N; combining independent measurements uses quadrature, and a 1/√N scaling sets the precision floor.

Significant figures and uncertainty reporting: quote the uncertainty to one or two significant digits and match the value's last digit to it.

Experimental design: randomize and control variables, calibrate instruments against known standards, and distinguish systematic bias from random scatter.

Least-squares fitting: slope uncertainty scales inversely with the spread of x; χ² per degree of freedom near 1 indicates a consistent model and error estimate.

Dimensional analysis: every physical equation must have identical dimensions on both sides; use [M][L][T] exponents to check or derive relations.

SI base units: meter, kilogram, second, ampere, kelvin, mole, candela; derived units are products of these (1 N = 1 kg·m·s^-2, 1 J = 1 N·m).

Waves: v = fλ; for a wave on a string v = sqrt(T/μ), for light in a medium v = c/n.

Superposition and interference: path difference Δr = mλ gives constructive, Δr = (m+1/2)λ gives destructive interference.

Doppler shift for sound: f' = f (v ± v_o)/(v ∓ v_s), with signs set by motion toward or away from the observer.

Snell's law: n1 sin θ1 = n2 sin θ2; total internal reflection occurs when θ1 exceeds θ_c = arcsin(n2/n1) for n1 > n2.

Photoelectric effect: K_max = hf − φ, with threshold frequency f_0 = φ/h; no emission below f_0 regardless of intensity.

Uncertainty propagation: for independent random errors, fractional errors add in quadrature; for z = x^a y^b, (σ_z/z)^2 = a^2(σ_x/x)^2 + b^2(σ_y/y)^2.

Newton's second law in inertial frames: F_net = dp/dt = ma for constant mass; in non-inertial frames add fictitious forces (−ma_frame).

Work–energy theorem: W_net = ΔK, with K = ½mv²; for conservative forces define U by F = −∇U and total E = K + U is conserved.

Momentum conservation holds when net external force is zero; collisions conserve momentum always, kinetic energy only if elastic.

Rotational dynamics: τ = Iα, L = Iω; for a rigid body about a fixed axis, I = Σ m_i r_i², and parallel-axis theorem I = I_cm + Md².

Angular momentum is conserved when net external torque vanishes; a spinning skater's ω scales as 1/I.

Simple harmonic motion: ẍ = −ω²x gives x = A cos(ωt + φ) with ω = sqrt(k/m) for a spring and ω = sqrt(g/L) for a small-angle pendulum.

Central-force orbits: effective potential U_eff = U(r) + L²/(2mr²); circular orbit requires dU_eff/dr = 0.

Kepler's third law for a two-body system: T² = 4π²a³/(G(M+m)); for planets about the Sun, T² ∝ a³.

Lagrangian mechanics: L = T − U, and the Euler–Lagrange equations d/dt(∂L/∂q̇) − ∂L/∂q = 0 generate the equations of motion.

Coulomb's law: F = k q1 q2 / r² with k = 1/(4πε0) ≈ 8.99×10^9 N·m²·C^-2; force is along the line joining charges.

Gauss's law: ∮E·dA = Q_enc/ε0; use symmetry (spherical, cylindrical, planar) to extract E for highly symmetric charge distributions.

Electric potential: V = −∫E·dl and E = −∇V; equipotential surfaces are perpendicular to field lines.

Capacitance: C = Q/V; parallel plate C = ε0 A/d, energy stored U = ½CV² = Q²/(2C); dielectrics multiply C by κ.

Lorentz force: F = q(E + v×B); magnetic forces do no work, so they change direction but not speed of a charged particle.

Biot–Savart and Ampère: ∮B·dl = μ0 I_enc; a long straight wire gives B = μ0 I/(2πr).

Faraday's law: EMF = −dΦ_B/dt; Lenz's law fixes the induced current direction to oppose the change in flux.

Maxwell's equations in vacuum predict EM waves with speed c = 1/sqrt(μ0 ε0); E, B, and k are mutually perpendicular with E = cB.

RC circuits: charging q(t) = Q(1 − e^(−t/RC)) with time constant τ = RC; RL circuits have τ = L/R.

de Broglie relation: λ = h/p; photon energy E = hf = hc/λ with hc ≈ 1240 eV·nm.

Heisenberg uncertainty: Δx Δp ≥ ħ/2 and ΔE Δt ≥ ħ/2; these are statistical spreads, not measurement disturbances.

Schrödinger equation: iħ ∂ψ/∂t = Ĥψ; stationary states satisfy Ĥψ = Eψ with |ψ|² the probability density.

Infinite square well of width L: E_n = n²π²ħ²/(2mL²), ψ_n = sqrt(2/L) sin(nπx/L); nodes increase with n.

Harmonic oscillator: E_n = (n + ½)ħω; zero-point energy ½ħω is nonzero, unlike the classical minimum.

Hydrogen atom: E_n = −13.6 eV/n²; orbital angular momentum magnitude L = sqrt(l(l+1))ħ with l < n.

Commutation and measurement: [x, p] = iħ; observables commute iff they can be simultaneously diagonalized and measured to arbitrary precision.

Identical particles: fermions obey antisymmetric wavefunctions and the Pauli exclusion principle; bosons obey symmetric wavefunctions.

Tunneling: a particle penetrates a barrier of height V0 > E with transmission decaying roughly as exp(−2κa), κ = sqrt(2m(V0−E))/ħ.

First law: ΔU = Q − W, with W = ∫P dV the work done by the system; U is a state function, Q and W are not.

Ideal gas law: PV = nRT = Nk_B T; average translational kinetic energy per molecule = (3/2)k_B T.

Entropy: dS = δQ_rev/T; the second law states ΔS_total ≥ 0 for any process, with equality only for reversible ones.

Carnot efficiency: η = 1 − T_c/T_h for a reversible engine operating between those reservoirs; no engine exceeds it.

Maxwell–Boltzmann distribution: number density ∝ exp(−E/k_B T); the equipartition theorem assigns ½k_B T per quadratic degree of freedom.

Statistical definitions: S = k_B ln Ω (microcanonical); partition function Z = Σ exp(−E_i/k_B T) with F = −k_B T ln Z.

Fermi–Dirac and Bose–Einstein distributions reduce to Maxwell–Boltzmann in the dilute limit; chemical potential controls occupation.

Maxwell relations follow from equality of mixed partials of thermodynamic potentials (U, H, F, G); e.g., (∂T/∂V)_S = −(∂P/∂S)_V.

Phase equilibrium requires equal temperature, pressure, and chemical potential; the Clausius–Clapeyron relation gives dP/dT = L/(TΔv).

Linear algebra: a matrix is diagonalizable iff it has a complete set of eigenvectors; Hermitian matrices have real eigenvalues and orthogonal eigenvectors.

Taylor expansion: f(x) = f(a) + f'(a)(x−a) + ½f''(a)(x−a)² + …; small-angle approximations sin θ ≈ θ and cos θ ≈ 1 − θ²/2 follow from it.

Fourier transform pair: f(t) = (1/2π)∫F(ω)e^(iωt)dω and F(ω) = ∫f(t)e^(−iωt)dt; narrow in time implies broad in frequency.

Vector calculus identities: ∇·(∇×A) = 0 and ∇×(∇f) = 0; the divergence and Stokes theorems convert between volume/surface and surface/line integrals.

Ordinary differential equations: a linear second-order ODE with constant coefficients is solved by the characteristic equation; damping regimes depend on the discriminant.

Error analysis: standard error of the mean is σ/√N; combining independent measurements uses quadrature, and a 1/√N scaling sets the precision floor.

Significant figures and uncertainty reporting: quote the uncertainty to one or two significant digits and match the value's last digit to it.

Experimental design: randomize and control variables, calibrate instruments against known standards, and distinguish systematic bias from random scatter.

Least-squares fitting: slope uncertainty scales inversely with the spread of x; χ² per degree of freedom near 1 indicates a consistent model and error estimate.