A collective pairing state generates a self-consistent gap

In the simplest spin-singlet model, the variational ground state is a product over pair labels of coherent empty and doubly occupied components. The amplitudes differ from one momentum to another, but their pairing phases are aligned in the chosen mean-field description. This is not a gas of pre-existing compact molecules: in weak coupling the characteristic pair size can greatly exceed the interparticle spacing. Particle-number projection can restore a definite total number without eliminating the thermodynamic pairing correlations captured by the mean-field construction.

Take an attractive matrix element −G in a symmetric energy shell |ξ|<Ec around the chemical potential. Let N₀ be the single-spin density of states in the same box normalization as G, and define g=N₀G. Assume it is constant over the shell. The gap equation sums the anomalous pair amplitude of all these states, including a thermal reduction factor. The positive quasiparticle energy contains the square root of ξ squared plus gap magnitude squared, so the unknown gap appears inside its own consistency integral.

At zero temperature the hyperbolic tangent is one. Integrating gives an inverse hyperbolic sine, which becomes a logarithm when the gap is much smaller than the cutoff. The resulting gap is approximately 2Ec times exp(−1/g). At the transition, take the gap to zero inside the integral and keep thermal occupation factors. Evaluating that logarithmic integral supplies the numerical prefactor 2 exp(γE)/π, where γE is Euler’s constant. This is the mean-field temperature where the pairing amplitude vanishes.

Dividing the zero-temperature gap by the transition scale eliminates both the cutoff and the coupling in this ideal limit. The ratio 2Δ₀/(kBTc) is approximately 3.528. Its exponential differs from the two-added-particle Cooper binding, which scales as exp(−2/g) with the same DOS convention. The many-body state gains coherence from reorganizing particles and holes around the Fermi surface. Using a total two-spin DOS in a formula derived for N₀ without adjusting the coupling would produce an erroneous factor in the exponent.

Strong-coupling retardation corrections, anisotropic gaps, multiple bands, pair breaking, or an energy-dependent density of states can change the gap ratio. In low-stiffness or low-dimensional systems, pairing correlations can persist above the temperature at which global phase coherence is established. The BCS mean-field Tc should therefore be identified with an observed transition only when phase fluctuations and other corrections are adequately controlled. A measured ratio different from 3.528 is evidence to examine the model, not by itself a unique diagnosis of an unconventional pairing mechanism.

The quantum lab represents one empty-versus-paired variational factor geometrically. Reset and set both controls to θ=90 degrees and φ=0 degrees, predicting equal empty and paired basis weights. Reset and set θ=60 degrees and φ=0 degrees, predicting paired weight one quarter if basis state one is assigned to the paired component. These are independent pure-state preparations, not a thermal scan or a self-consistent solution. Playback changes relative phase while leaving populations fixed. A complete BCS state needs many such factors linked through the common gap equation.