Foundations

For a real short-range central potential, each elastic orbital partial wave has S_l=exp(2iδ_l), |S_l|=1.

For the stated real short-range elastic central-potential problem, the integrated elastic cross section equals σ_el=(4π/k)Im f(0).

For short-range three-dimensional elastic central scattering without a threshold singularity, the s-wave scattering length is a_s=−lim_(k→0)tanδ₀/k.

Vacuum fields E=−∇φ−∂_tA and B=∇×A are unchanged by A′=A+∇χ, φ′=φ−∂_tχ.

In Coulomb gauge ∇·A=0, the vacuum scalar potential satisfies ∇²φ=−ρ/ε₀ while the vector potential obeys (∇²−c⁻²∂_t²)A=−μ₀J_T, J_T=J−ε₀∇∂_tφ and ∇·J_T=0.

The source-generated vacuum Lorenz potentials are φ(r,t)=(1/(4πε₀))∫ρ(r′,t−R/c)d³r′/R and A(r,t)=(μ₀/(4π))∫J(r′,t−R/c)d³r′/R, where R=|r−r′|.

Differentiating the causal potentials gives E=(1/(4πε₀))∫([ρ]n/R²+[ρ̇]n/(cR)−[J̇]/(c²R))d³r′ and B=(μ₀/(4π))∫([J]/R²+[J̇]/(cR))×n d³r′.

For a localized steady conserved current, the magnetic dipole moment is m=(1/2)∫r′×J(r′)d³r′.

A fixed magnetic moment in a spatially uniform external field experiences torque τ=m×B.

A localized nonrelativistic electric dipole radiates E_rad≈n×(n×p̈(t−r/c))/(4πε₀c²r) and B_rad=n×E_rad/c.

For a homogeneous nonmagnetic ohmic medium with real ε and σ, a wave exp(ikz−iωt) satisfies k²=μ₀εω²+iμ₀σω.

In a cold collisionless unmagnetized electron plasma with fixed neutralizing ions, transverse waves obey ω²=ω_p²+c²k² and ω_p²=n_e e²/(m_eε₀).

For a differentiable real model Y=f(X), first-order output covariance is Σ_Y≈JΣ_XJᵀ, J_ai=∂f_a/∂x_i at the input estimate; the rule is exact for an affine model with finite input covariance.

In a constructed scalar model Y=X² with X uniformly distributed on [−a,a], a>0, exact distribution propagation gives E(Y)=a²/3 and Var(Y)=4a⁴/45, hence u(Y)=2a²/√45.

For a validated linear first-order sensor model τẏ+y=Kx with τ>0, a step to x_f gives y(t)=Kx_f+[y(0)−Kx_f]e^(−t/τ).

A nonrelativistic point-Coulomb two-body bound state separates as ψ_nlm=R_nl(r)Y_lm(θ,φ).

For two orthonormal states with H₀=diag(E₁,E₂), E₂>E₁, and off-diagonal drive H₁₂=H₂₁*=ħΩ exp(iωt)/2, the initial state |1⟩ has exact probability P₂(t)=Ω² sin²(t√(Ω²+δ²)/2)/(Ω²+δ²), δ=ω−(E₂−E₁)/ħ, and P₁=1−P₂.

A liquid-drop binding parameterization combines a positive volume term with surface, proton Coulomb and neutron–proton asymmetry penalties, plus an even–odd pairing term: B≈a_vA−a_sA²ᐟ³−a_cZ(Z−1)A⁻¹ᐟ³−a_a(A−2Z)²/A+δa_pA⁻³ᐟ⁴.

Removing one neutron or proton from a specified ground-state nucleus costs S_n=B(A,Z)−B(A−1,Z) or S_p=B(A,Z)−B(A−1,Z−1).

A rotationally invariant central nucleon-pair model can include V(r)=V_c(r)+V_s(r)S_p·S_n/ħ².

In a spherical nuclear mean field, single-nucleon states fill separately for protons and neutrons with 2j+1 states in each j level.

An alpha channel changes (A,Z) to (A−4,Z−2) plus ⁴He.

Weak beta channels include n→p+e⁻+ν̄_e, p→n+e⁺+ν_e, and bound-electron capture p+e⁻→n+ν_e within a nucleus.

For an electromagnetic nuclear transition with good angular momentum and parity, a single emitted photon has multipolarity L≥1 satisfying |J_i−J_f|≤L≤J_i+J_f.

Independent constant decay channels have total rate λ=Σ_iλ_i and branching fractions b_i=λ_i/λ.

For projectile rest mass m_a and target m_A initially at rest, squared center-of-momentum energy is E_cm²=(m_a+m_A)²c⁴+2m_Ac²K_a.

For a nonrelativistic point charge scattered repulsively by a heavy fixed point charge with V(r)=κ/r, κ>0, impact parameter b=κ cot(θ/2)/(2E).

Photoelectric absorption transfers a photon to a bound electron with ion recoil/relaxation; an approximate emitted-electron kinetic energy is hν−B when recoil is negligible.

Stopping power S(E)=−dE/dℓ is the mean energy loss per path length in a specified material.

For a narrow monoenergetic photon beam through a homogeneous medium with constant removal coefficient μ, the uncollided intensity obeys I=I₀exp(−μℓ), with mean free path 1/μ.