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):

enter image description here

  • $\begingroup$ 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. $\endgroup$ – dgwp Nov 16 '17 at 20:35
  • $\begingroup$ @dgwp, check my new answer. $\endgroup$ – stafusa Nov 17 '17 at 1:54
  • $\begingroup$ 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 $l=0$! $\endgroup$ – dgwp Nov 17 '17 at 11:11
  • $\begingroup$ @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. $\endgroup$ – stafusa Nov 17 '17 at 12:55
  • $\begingroup$ 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? $\endgroup$ – dgwp Nov 17 '17 at 17:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.