Foundations

For repulsively charged nuclei with κ=Z_aZ_be²/(4πε₀)>0 and relative speed v, the Sommerfeld parameter is η=κ/(ħv) and the leading low-energy Coulomb penetration exponential is exp(−2πη)=exp[−√(E_G/E)], E_G=2μ(πκ/ħ)².

For a real periodic one-dimensional potential of period a, define V_G=a⁻¹∫₀ᵃV(x)e^(−iGx)dx, G=2π/a, and ε_k=ħ²k²/(2m).

For a periodic chain of N orthonormal localized orbitals, one per site with spacing a, take H_nm=ε₀δ_nm−t(δ_(n+1,m)+δ_(n−1,m)).

An ion's equilibrium potential balances its chemical and electrical driving terms.

For n integrable Hamiltonian degrees of freedom on a regular compact connected common level set of n independent commuting integrals, local canonical actions I_i and angles θ_i describe torus motion.

For H=H₀(I)+εH₁(I,θ), an integer resonance k·ω(I)=0 makes that Fourier mode unsuitable for ordinary nonresonant elimination.

Independent second-class constraints φ_a=0 have an invertible Poisson matrix C_ab={φ_a,φ_b}.

For separable H=pᵀM⁻¹p/2+V(q), fixed-step kick–drift–kick Verlet is second order, reversible and symplectic.

A positive-definite local energy difference can prove Lyapunov stability of an isolated equilibrium even when its quadratic Hessian is degenerate.

A finite-outcome POVM is a family of positive operators E_a whose sum is the identity.

An instrument associates outcome a with a completely positive trace-nonincreasing map ℐ_a.

A quantum channel Φ(ρ)=Σ_kK_kρK_k† with Σ_kK_k†K_k=I remains positive when extended by an identity map on an arbitrary ancilla.

A time-homogeneous completely positive trace-preserving semigroup has a GKSL generator: a Hamiltonian commutator plus dissipators with nonnegative rates.

For stationary zero-mean Gaussian phase noise f(t), a qubit coherence factor is exp[−½∫₀ᵀ∫₀ᵀK(t−t′)dt dt′], where K is the covariance of angular-frequency noise.

For a uniform dimensionless scalar order parameter m with symmetry m↦−m, an effective free-energy density f=f₀+a m²/2+b m⁴/4 has minima m=0 for a>0 and m²=−a/b for a<0 when b>0.

Connected order-parameter correlations define a correlation length ξ.

Coarse-graining changes couplings while preserving long-distance observables after rescaling.

In the local London model, ∇×j_s=−n_s q_s²B/m_s.

Ginzburg–Landau describes superconductivity through a complex order parameter, symmetry and a gradient/field-dependent free energy, yielding a coherence length and electromagnetic response.

For the local single-component GL model, the SI free-energy density includes α|ψ|²+β|ψ|⁴/2, a covariant gradient cost and B²/(2μ₀).

The homogeneous spin-singlet mean-field pairing model gives quasiparticles E_k=√(ξ_k²+|Δ_k|²), whose thermal populations enter a self-consistent gap equation.

Bardeen, Cooper and Schrieffer's 1957 paper proposed a phonon-mediated attractive interaction and a correlated paired ground state, connecting an excitation gap to thermodynamic and electromagnetic behaviour.

In correlated superconductors, spin-fluctuation-mediated pairing, electron–phonon contributions and other electronic correlations must be tested against gap symmetry, spectroscopy and thermodynamics for the particular material.

The asymptotically flat, spherical vacuum exterior has ds²=−(1−r_s/r)c²dt²+(1−r_s/r)⁻¹dr²+r²dΩ², r_s=2GM/c².

Two static clocks outside a Schwarzschild source compare proper rates through √(1−r_s/r).

Weak far-zone gravitational waves change proper separations through transverse-traceless strain.

For a differentiable local field Lagrangian, a continuous global symmetry that changes it by a total divergence yields an on-shell conserved Noether current.

LIGO's 2016 report of the 14 September 2015 event combined a coherent transient in two detectors with noise-background tests and waveform comparison.

Quantizing a real free scalar field replaces its amplitudes by operators with [φ(t,x),π(t,y)]=iδ³(x−y) in ħ=c=1 units.

The interaction-picture time-ordered expansion organizes scattering amplitudes in couplings; loops introduce regulated momentum integrals.