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Quantum Derivation of Ginzburg-Landau Equation. New Formula for Penetration Depth

Authors:	Shigeji Fujita Salvador Godoy

Affiliation:	1. Department of Physics, State University of New York at Buffalo, Buffalo, New York, 14260, USA 2. Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México, México, 04510, D. F., México

Abstract:	The Cooper pair (pairon) field operator ψ(r,t) changes in time, following Heisenberg’ s equation of motion. If the system Hamiltonian $mathcal{H}The Cooper pair (pairon) field operator ?(r,t) changes in time, following Heisenberg's equationof motion. If the system Hamiltonian contains the pairon kineticenergies h₀, the condensation energy per pairon(< 0) and the repulsive point-like potential(r₁ –r₂), > 0, the evolution equation for ?is non-linear, from which we obtain the Ginzburg-Landau equation: for the complex order parameter $Psi _sigma (r): = < r\|n^{1/2} \|sigma >$ , where denotes thestate of the condensed pairons, and n the pairon densityoperator. The total kinetic energy h₀ forelectron (1) and hole(2) pairons is $h_0 Psi _sigma (r) = left{ {frac{1}{2}v_F^{(1)} \| - ibar hnabla + 2eA(r)\| + frac{1}{2}v_F^{(2)} \| - ibar hnabla - 2eA(r)\|} right}Psi _sigma (r)$ where $v_F^{(j)} equiv (2varepsilon _F /m_j )^{1/2}$ are Fermi velocities, and A thevector potential. A new expression for the penetration depth isobtained: $lambda = frac{c}{e}left[ {frac{p}{{4pi n_0 (v_F^{(2)} + v_F^{(1)} )}}} right]^{1/2}$ where p and n₀ are respectively themomentum and density of condensed pairons.

Keywords:	Ginzburg-Landau equations penetration depth microscopic derivation of G-L equations validity of G-L equations
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