We usually define mean field order parameter (gap function) in BCS theory $$ \Delta(r_1,r_2)_{s,s'} = \langle GS | \hat{\psi}_s(r_1)\hat{\psi}_{s'}(r_2) |GS\rangle, $$ where $\hat{\psi}_s(r_1)$ is field operator of electron on position $r_1$ with spin $s$ and $|GS\rangle$ is ground state of a superconductor.
Can we understand this quantity as a wavefucntion of a Cooper pair just as solution of two electrons interacting with each other via an attractive potential?
I don't know my answer make sense or not what i understand about that order parameters given by
wave function mod square it represents the copper pair forming probability density which is zero above some critical temperature and critical fields.
Now the formation copper pair is little bit tricker but this is how i imagine with classical pictures. Imagine +ve charge uniform density lattice . Now imagine two electron moving parallel to each other one of them slightly behind other. Since electron is negatively charge it will attract near by +ve core resulting a compression like mesh we call that creation of phonon represents as 'psi+' opperator acted on vacuum. Due to that a higher dense +ve charge density occurs resulting attractive force on electron slightly behind it. Now the oscillation is in sound velocity range while electron moving much faster than sound . Even electron not their the phonon created will be gonna continue their resulting attracting in other electron thats electron phonon electron interaction make the Cooper pair possible
