GENERALIZATION OF LONDONS' EQUATIONS BASED ON THE WAVE-FUNCTION OF THE COOPER'S PAIRS

The first phenomenological theory for the superconductivity consists of two Londons' equations. After Londons' theory, Bardeen, Cooper, and Schrieffer set up the microscopic theory (BCS theory) for conventional superconductors. The BCS theory states that, below the critical temperature, electrons of a metal are pairing to form quasi-particles, named as Cooper's pairs. The supperconducting state is the condensate of macroscopic number of Cooper's pairs in a unique quantum state. We give a detail derivation of Londons' equations based on the BCS picture, emphasizing the local gauge invariant and taking the variation of the density of Cooper's pairs into account.