Proof of the phase coherence in the Bardeen-Cooper-Schrieffer theory of superconductivity from first principles

Even though the phase coherence is a basic principle of the Bardeen-Cooper-Schrieffer theory of superconductivity, nowhere in the literature can a direct and simple proof of it be found. Here it is shown that a minimization with respect to the phases of the probability amplitudes of the superconducting wavefunction of the theory leads naturally to phase coherence in a simple and straightforward way. The proof is given in terms of first principles and the uniqueness of the phase coherence solution is shown. This paper aims to provide a more physical insight into the origin of a concept that it is widely accepted in the above theory.