Cluster Decomposition Principle and Two-Electron Wave Function of the Cooper Pair in the BCS Superconducting State

TL;DR

This paper derives the explicit form of the two-electron wave function in BCS superconductors, confirming the cluster decomposition principle and identifying the eigenfunction as a Cooper pair with a size matching Pippard's coherence length.

Contribution

It provides an explicit form of the maximum eigenfunction of the RDM2 in BCS states and establishes its physical interpretation as a Cooper pair.

Findings

Cluster decomposition principle holds in BCS superconductors.

Eigenfunction is a spin singlet and isotropic.

Electron pair size matches Pippard's coherence length.

Abstract

We present the explicit forms of the maximum eigenvalue and the corresponding eigenfunction for the second-order reduced density matrix (RDM2) of the BCS superconducting state (SS). Using these quantities, we deal with two topics in the present paper. As the first topic, it is shown that the cluster decomposition principle holds in the BCS-SS. This proof gives a theoretical foundation that the abnormal density can be chosen as the order parameter of the SS. As the second topic, it is shown that such an eigenfunction is spin singlet and spatially extends isotopically, and further that the mean distance of two electrons which consists of the above eigenfunction is in a good agreement with Pippard's coherence length. This means that maximum geminal of the RDM2 of the BCS-SS can be regarded as the Cooper pair itself which are condensed to the same energy level in a number of O(N).