Foundations

For a narrow-band wave packet on a smooth real dispersion branch, its leading envelope velocity is v_g=dω/dk at the carrier, while phase velocity is ω/k.

For V(x)=−V₀ inside |x|<a/2 and zero outside, bound states have −V₀<E<0.

For a rotationally invariant scalar potential V(r), simultaneous orbital-angular-momentum eigenstates separate as ψ=R_l(r)Y_lm(θ,φ).

At weak magnetic field and T≈0, a noninteracting spin-½ electron gas has spin magnetization M=μ_B²g(E_F)B and susceptibility χ_P=μ₀μ_B²g(E_F).

For a rotationally symmetric localized electronic orbital, the quadratic magnetic-field term in the nonrelativistic Hamiltonian gives δE=e²B²⟨r²⟩/(12m_e).

A smooth type-2 generating function F₂(q,P,t) defines p_i=∂F₂/∂q_i and Q_i=∂F₂/∂P_i.

Hamilton's principal function satisfies ∂_tS+H(q,∂_qS,t)=0.

For one regular periodic Hamiltonian orbit, I=(1/2π)∮p dq and its conjugate angle θ advances at ω=∂H/∂I; equivalently dI/dE=T/(2π).

For a smooth classical fluid with mass continuity, material acceleration is D_tv=∂_tv+(v·∇)v and inviscid momentum obeys ρD_tv=−∇P+ρg.

For a smooth inviscid, nonconducting fixed-composition fluid, D_ts=0 and ρD_te=−P∇·v, where e and s are internal energy and entropy per mass.

Linearizing mass continuity and inviscid momentum around a uniform stationary fluid with δP=c_s²δρ gives ∂_t²δP=c_s²∇²δP.

For an autonomous smooth system ẋ=f(x), equilibria satisfy f(x*)=0 and nearby perturbations obey δẋ≈Jδx, J_ij=∂f_i/∂x_j at x*.

A bifurcation is a qualitative change in a parameterized dynamical system.

At an interface between lossless isotropic nonmagnetic dielectrics, normal energy-flux ratios are R_s=[(n₁cosθ_i−n₂cosθ_t)/(n₁cosθ_i+n₂cosθ_t)]² and R_p=[(n₂cosθ_i−n₁cosθ_t)/(n₂cosθ_i+n₁cosθ_t)]², with T_s,p=1−R_s,p.

An isolated Newtonian gravitational binary has relative semimajor axis a=a₁+a₂ and period T satisfying T²=4π²a³/[G(m₁+m₂)].

For a wave packet in one smooth band E(k), neglecting anomalous velocity, v=(1/ħ)∇_kE and ħk̇=F.

In nondegenerate three-dimensional parabolic bands, n=N_c exp[−(E_c−μ)/(k_BT)] and p=N_v exp[−(μ−E_v)/(k_BT)], where N_c,v=2(2πm_c,v*k_BT/h²)³ᐟ² for one isotropic valley including two spins.

At weak electric field in a two-carrier relaxation-time band model, v_n=−μ_nE and v_p=+μ_pE with positive mobilities μ=|e|τ/m*.

Away from turning points, a slowly varying one-dimensional potential admits leading stationary WKB waves ψ≈[A e^(i∫k dx)+B e^(−i∫k dx)]/√k in an allowed region and ψ≈[C e^(∫κ dx)+D e^(−∫κ dx)]/√κ in a forbidden region, with k=√[2m(E−V)]/ħ and κ=√[2m(V−E)]/ħ.

At a simple turning point x_t with V′(x_t)>0, an exterior wave decaying to the right, C exp[−∫_(x_t)^x κ dx]/√κ, determines the leading allowed-left wave 2C cos[∫_x^(x_t)k dx−π/4]/√k.

For a one-dimensional smooth well with two separated simple turning points x₁<x₂ and decaying tails on both sides, leading WKB matching selects ∫_(x₁)^(x₂)√[2m(E_n−V)]dx≈πħ(n+½), n=0,1,… .

For a smooth opaque barrier between two simple turning points, the leading transmitted-to-incident probability-flux fraction is T≈exp[−2∫_(x₁)^(x₂)κ(x)dx], κ=√[2m(V−E)]/ħ.

For an electron in a central electrostatic potential energy V(r), the leading low-velocity corrections to p²/(2m_e)+V are H_rel=−p⁴/(8m_e³c²), H_SO=V′(r)L·S/(2m_e²c²r) and H_D=ħ²∇²V/(8m_e²c²).

For a spin-one-half electron bound to an infinitely heavy point nucleus of charge +Ze, the binding energy through order (Zα)⁴ is E_nj≈−m_ec²(Zα)²/(2n²)−m_ec²(Zα)⁴[n/(j+½)−¾]/(2n⁴), with α=e²/(4πε₀ħc).

At leading electron g_s=2, a uniform magnetic field along z adds H_B=μ_BB(L_z+2S_z)/ħ, μ_B=eħ/(2m_e).

For a fixed smooth finite-dimensional Hamiltonian protocol H(s), s=t/T, an initially isolated nondegenerate eigenstate with a nonzero gap throughout the path follows its instantaneous eigenspace in the sufficiently slow limit.

For an isolated normalized eigenstate taken adiabatically around a closed parameter loop C, the geometric phase is γ_n(C)=∮_C A_n(R)·dR modulo 2π, with A_n=i⟨n(R)|∇_R n(R)⟩.

A nonrelativistic spinless particle of signed charge q in electromagnetic potentials has H=(−iħ∇−qA)²/(2m)+qφ and kinetic momentum Π=p−qA.

For stationary three-dimensional scattering by a short-range real potential, a unit-amplitude incident plane wave has asymptotic form ψ≈exp(ikz)+f(θ,φ)exp(ikr)/r.

Replacing the exact scattering wave inside the potential integral by the incident plane wave gives f_B(k′,k)=−m∫exp[i(k−k′)·r]V(r)d³r/(2πħ²).