# Spherical symmetry of Cooper pair wave function

Can someone please explain to me how the wave function of a Cooper pair is spherically symmetric?

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):

• Sorry, but it is not clear at all from that page. I have not found a book that explains it properly either, which makes me think that some authors don't really understand it. I would like to see a step by step explanation, starting from the explicit form of the wavefunction. – dgwp Nov 16 '17 at 20:35
• @dgwp, check my new answer. – stafusa Nov 17 '17 at 1:54
• Thank you very much. Do we have to assume that the angular momentum is zero then? The sketchy explanations I have seen in textbooks seem to write the $e^{i\mathbf{k}\cdot\mathbf{r}}$ as a plane wave expansion, then integrate (rather than sum) over $\mathbf{k}$ to get something like $\int\;dk\sin(kr)/k$, but I don't know what has happened to the anguar dependence. And it seems that they have already assumed that $l=0$! – dgwp Nov 17 '17 at 11:11
• @dgwp, I think some authors are indeed assuming the symmetry from non-rigorous arguments, while others are solving Schrödinger's equation in 1D, but the best argument is that the lowest order of the expansion in harmonics is spherically symmetrical. – stafusa Nov 17 '17 at 12:55
• So, if I understand correctly, you are saying that the wavefunction for the relative motion of the Cooper pair contains the ground state and higher-energy states within a single wavefunction? – dgwp Nov 17 '17 at 17:04