Spherical symmetry of Cooper pair wave function Can someone please explain to me how the wave function of a Cooper pair is spherically symmetric?
 A: 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.


[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
$$ \Psi(r) = \cos{(k_F r)} \,K_0(r/\pi\xi_0), $$
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):

