I was reading Lectures on BCS theory by Prof. Rafael M. Fernandes (https://portal.ifi.unicamp.br/images/files/graduacao/aulas-on-line/fen-emerg/lecture_notes_BCS.pdf). On page 4 of text, equation 16 which is $$\Big( -\frac{\hbar^2\nabla^2_{r_1}}{2m} - \frac{\hbar^2\nabla^2_{r_2}}{2m} + V(r_1-r_2) \Big) \psi (r_1,r_2)=E\psi (r_1,r_2)$$ does not include Coulomb repulsion between electrons. The potential mentioned is attractive in nature. I do not understand why they have not taken Coulomb repulsion in account?
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The calculation in question is meant to show that if the electrons have a net attractive interaction - no matter how small - then Cooper pairs will form at the Fermi surface. There is no mention (at the moment) of how an attractive interaction might arise in the first place.
In reality, at least in conventional superconductors, this attraction arises because the electrons couple to vibrations of the positively-charged lattice. Under appropriate conditions (i.e. at sufficiently low temperature), this effective interaction between electrons (together with screening effects) is more important than the electrostatic repulsion between them, and Cooper pairs will form.
answered Apr 13, 2020 at 19:25
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$\begingroup$ So, you are saying for net interactions between the electrons to be attractive, the phonon-electron interaction must be greater than Coulomb repulsion? $\endgroup$ Apr 14, 2020 at 0:02
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$\begingroup$ @KartikChhajed Well, the screened Coulomb repulsion, but yes that's correct. And again, this is a specific model - in more exotic superconductors, other mechanisms may create a net attraction between electrons, but the precise mechanism is unimportant for this discussion. $\endgroup$ Apr 14, 2020 at 0:47