Foundations

Quantum angular momentum obeys [J_i,J_j]=iħ ε_ijk J_k.

Independent commuting angular momenta combine as J=J₁+J₂.

For an isolated nondegenerate eigenvalue of H₀ and a small Hermitian perturbation V, the leading energy corrections are ⟨n|V|n⟩ and Σ_{k≠n}|⟨k|V|n⟩|²/(E_n⁰−E_k⁰).

Inside a degenerate eigenspace of H₀, first-order energy shifts and appropriate zeroth-order states come from diagonalizing the projected Hermitian perturbation PVP.

Starting in eigenstate i of fixed H₀, a weak perturbation V(t) gives first-order amplitude c_f(t)=−(i/ħ)∫_{t₀}^t V_fi(t′)exp[i(E_f−E_i)(t′−t₀)/ħ]dt′ for f≠i.

In the long-wavelength electric-dipole approximation, atomic transition strength depends on the dipole matrix element.

For a self-adjoint Hamiltonian bounded below, every normalized trial state with a finite energy expectation satisfies ⟨ψ_trial|H|ψ_trial⟩≥E₀.

For a simple compressible equilibrium system, U(S,V,{N_i}) has differential dU=T dS−p dV+Σ_i μ_i dN_i.

The Legendre transforms H=U+pV and F=U−TS change the natural state variables: dH=T dS+V dp and dF=−S dT−p dV at fixed composition.

Equality of mixed derivatives of smooth equilibrium potentials gives (∂T/∂V)_S=−(∂p/∂S)_V, (∂T/∂p)_S=(∂V/∂S)_p, (∂S/∂V)_T=(∂p/∂T)_V and (∂S/∂p)_T=−(∂V/∂T)_p.

For a fixed amount of simple compressible matter, C_V=(∂U/∂T)_V and C_p=(∂H/∂T)_p.

A cyclic heat engine drawing heat Q_h from a reservoir T_h and rejecting Q_c to T_c delivers W=Q_h−Q_c.

Thermal equilibrium is transitive, permitting a common temperature scale.

For an isolated equilibrium system with fixed energy shell, volume and particle number, the microcanonical model gives equal probability to accessible microstates.

For canonical Z=Σ_i exp(−βE_i), F=−k_BT lnZ, U=−∂_β lnZ and S=k_B(lnZ+βU).

For a system exchanging energy and particles with reservoirs at T and μ, p_i,N=exp[−β(E_i,N−μN)]/Ξ and Ξ=Σ_N,i exp[−β(E_i,N−μN)].

In a classical canonical model, each independent quadratic term in an unrestricted coordinate or momentum contributes k_BT/2 to the mean energy.

For a dilute classical ideal gas in its rest frame, the probability density of a particle velocity is (m/(2πk_BT))³ᐟ²exp[−m|v|²/(2k_BT)].

Pure phases coexist at common T and p when their molar Gibbs energies are equal.

Thermal radiation in a vacuum cavity at temperature T has spectral energy density per angular frequency u_ω=ħω³/[π²c³(exp(ħω/(k_BT))−1)].

In vacuum electrostatics, a point charge q at r′ produces E(r)=q(r−r′)/(4πε₀|r−r′|³).

Electrostatic potential obeys ∇²φ=−ρ/ε₀ in vacuum; charge-free regions obey Laplace's equation.

For a charge q at r₀ outside a grounded conducting sphere of radius a centered at the origin, an image q′=−qa/|r₀| at r′=a²r₀/|r₀|² gives the exterior potential q/(4πε₀|r−r₀|)+q′/(4πε₀|r−r′|).

Outside a localized charge distribution, the potential expands in powers of inverse distance.

A localized steady current distribution in vacuum produces B(r)=μ₀/(4π)∫J(r′)×(r−r′)/|r−r′|³d³r′.

Macroscopic electric polarization P gives bound volume charge ρ_b=−∇·P and surface charge σ_b=P·n at a material boundary.

Magnetization M contributes bound volume current J_b=∇×M and surface current K_b=M×n.

For a stationary interface with unit normal n from medium 1 to 2, n·(D₂−D₁)=σ_free, n·(B₂−B₁)=0, n×(E₂−E₁)=0 and n×(H₂−H₁)=K_free.

In a homogeneous nonmagnetic isotropic dielectric, a monochromatic plane wave has k²=ε_r(ω)ω²/c² and B₀=k×E₀/ω.

A damped bound-electron oscillator gives ε_r(ω)=1+Ne²/[ε₀m(ω₀²−ω²−iγω)] for the exp(−iωt) convention.