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I am learning about BCS theory and how Cooper pairs are formed. They have opposite $k$ and opposite spin. But what I do not understand is why Cooper pairs don't break up on their own. I have pictures like this to get an idea:

picture shows attractive interaction between two electrons within a crystal lattice

But if one electron has $k,$ and the other electron has a momentum $-k,$ does this mean they move in different directions in real space? How can the interaction be attractive if so?

It seems like I am missing something here. Maybe you can explain how Cooper pairs can move in the same direction but have opposite momentum $k$?

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The k, -k combination is for a Cooper pair at rest. If it moves it will be k+q, -k+q.

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