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The canonical momentum of the charge carriers in a superconductor can be set to zero, and this may be used to derive the London equations.

However, why can can we set the canonical momentum to zero? I am struggling to find any sources of justification...

Ideally answers would contain both theoretical and experimental perspectives. Also, how is this related to the actual momentum of the Cooper pair?

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You might want to read Mahan's Many particle physics, chap. 10.1 on the Cooper instability.

The binding of electrons into a Cooper pair is a purely statistical effect that comes from the Fermi distribution at zero-T. The binding energy can be arbitrarily small ; what makes the electrons pair is the sharpness of the Fermi-Dirac function close to the Fermi energy. In short, you need electrons to pair with opposite momentum because you need the energy of relative motion to be located exactly at the discontinuity of momentum distribution (which is located at the Fermi energy) to cause the binding of electrons.

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