Quantum Statistical Derivation of a Ginzburg-Landau Equation

Abstract

The pairon field operator ψ ( r , t ) evolves, following Heisenberg’s equation of motion. If the Hamiltonian H contains a condensation energy α 0 (<0) and a repulsive point-like interparticle interaction , , the evolution equation for ψ is non-linear, from which we derive the Ginzburg-Landau (GL) equation: for the GL wave function where σ denotes the state of the condensed Cooper pairs (pairons), and n the pairon density operator ( u and are kind of square root density operators). The GL equation with holds for all temperatures ( T ) below the critical temperature T c , where ε g ( T ) is the T -dependent pairon energy gap. Its solution yields the condensed pairon density . The T -dependence of the expansion parameters near T c obtained by GL: constant is confirmed.