Spherical symmetry of Cooper pair wave function

Summary: There's an attraction, mediated by the material lattice vibrations (phonons), between electrons. Considering that this attractive potential depends only on the distance $r$ between the electrons, we have an atomic like situation, where the lowest energy level is spherically symmetrical.

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[comment] I would like to see a step by step explanation, starting from the explicit form of the wave-function

According to Kadin's very didactic Spatial Structure of the Cooper Pair (arXiv), a wave-function for the pair is

$$ Psi(r) = cos{(k_F r)} ,K_0(r/pixi_0), $$ where $k_F$ is the "Fermi wave-vector at the top of the Fermi see", and s the zeroth-order modified Bessel function. Since $Psi$ depends only on the radial distance, the wave-function is spherically symmetrical.

Note that that's the s-wave and that

there has been considerable discussion of d-wave pairing symmetry as applied to the cuprates [7], as well as p-wave symmetry in ruthenates [8]. It is straightforward to modify the spherically symmetric quasi-atomic orbital to include one or more angular nodes

Edkins' PhD. thesis (mirror) describes a similar picture (p. 7):

In the simplest case, the wave-function of the Cooper-pair can be written as a product of orbital and spin parts. [...] If we were to expand the orbital wave-function in spherical harmonics (which is valid in free space), spin-singlet pairs will have angular momentum quantum number $l=0,2,ldots$, which we call s- and d-wave respectively, by analogy with the orbitals of the hydrogen atom. Likewise, spin-triplet superconductors will have $l=1,3,ldots$ corresponding to p- and f-wave respectively.

$Longrightarrow$ And this description is not only valid for free space, since, according to Fossheim and Sudboe Superconductivity: Physics and Applications (my emphasis):

The lowest order square lattice harmonics [...] have common properties with the lowest spherical harmonics functions that are the basis functions for the isotropic case.

And there's also Kai Hock's explanation (pdf):