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After studying the BCS theory of superconductivity, I do not understand this quote from my notes:

"The phase of the BCS wavefunction is well defined."

I do not see this "phase" from the coherent state form of the BCS ground state. What is it?

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It just means that in the BCS ground state, all the electrons form bosonic pair states that condense in a single wavefunction, which hase a phase $\Phi_0$. In this case you can say that all the electrons "share" the same phase $\Phi_0$.

This is not the case in a general metallic state where electrons are much less correlated : their relative phases are random. In this case you say that the "global phase" is not well-defined.

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