Quantum Statistical Derivation of a Ginzburg-Landau Equation

Shigeji Fujita,Akira Suzuki

doi:10.4236/jmp.2014.516157

Abstract

The pairon field operator ψ(r,t) evolves, following Heisenberg’s equation of motion. If the Hamiltonian H contains a condensation energy α0(<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 σdenotes 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 Tc, where eg(T) is the T-dependent pairon energy gap. Its solution yields the condensed pairon density . The T-dependence of the expansion parameters near Tc obtained by GL: constant is confirmed.

Highlights

In 1950 Ginzburg and Landau (GL) [1] proposed a revolutionary idea that below the critical temperature Tc aHow to cite this paper: Fujita, S. and Suzuki, A. (2014) Quantum Statistical Derivation of a Ginzburg-Landau Equation
We derive the GL wave equation microscopically and show that the GL equation is valid for all temperature below Tc
In the derivation we found that the particles that are described by the GL wave function Ψ (r ) must be bosons

Summary

Introduction

In 1950 Ginzburg and Landau (GL) [1] proposed a revolutionary idea that below the critical temperature Tc a. We derive the GL wave equation microscopically and show that the GL equation is valid for all temperature below Tc. Fujita, Ito and Godoy in their book [5] discussed the moving pairons. The energy wq increases linearly with momentum q(= q ) for small q This behavior arises since the pairon density of states is strongly reduced with increasing momentum q and dominates the q2-increase of the kinetic energy. This linear dispersion relation means that a pairon moves like a massless particle with a common speed vF 2.

Ginzburg-Landau Equation at 0 K

Discussion

Conclusions