Quantum Statistical Derivation of the Ginzburg–Landau Equation.

The Cooper pair (pairon) field operator ψ†(r,t) changes, following Heisenberg's equation of motion. If the Hamiltonian ℋ contains pairon kinetic energies h0, a condensation energy α(<0) and a repulsive point-like interpairon interaction βδ(r1-r2), β>0, the evolution equation for ψ is nonlinear, from which we obtain the Ginzburg–Landau (GL) equation: [Formula: see text] for the GL wave function Ψσ(r)≡ <r| n1/2|σ>, where σ denotes the state of the condensed pairons, and n the density operator. The GL equation with α=-εg(T) is shown to hold for all temperatures (T) below Tc, where εg is the pairon energy gap. Its equilibrium solution yields that the condensed pairon density n0(T)=|Ψσ(r)|2 is proportional to εg(T). The original GL T-dependence of the expansion parameters near Tc:α=-b(Tc-T), β= constant is justified. With the assumption of h0, a new formula for the penetration depth is obtained.