$\frac{1}{\lvert\ \vec(r_{1})-\vec(r_{2})\rvert}$ is a hermitian operator so I am supposing there could potentially be an observable with this which might be experimentally measured. We can measure the energy difference spectroscopically between the singlet and triplet states, but that doesn't tell us how much of this energy difference is due to a change in electron-electron repulsion versus how much is due to a change in interaction energies of the electrons with the nucleus. Measuring the electron-electron repulsive potential energy directly would give a way of indirectly measuring the average distance between them.

  • $\begingroup$ I don't see why that would be relevant. I mean to measure that quantity. You have to think about position as a macroscopic quantity, you can't really define it at a quantum level. $\endgroup$ – iharob Nov 27 '15 at 17:03
  • $\begingroup$ Maybe it could provide clues to getting exact solution to Schroedinger's equation for Helium atom. $\endgroup$ – dualredlaugh Nov 27 '15 at 19:15
  • $\begingroup$ I don't think it's solvable, at least not with any of the traditional approaches, back when I first learned about Schrodinge's equation I became an expert in it. And I actually tried to solve the He, and I failed of course. You can however use numerical methods and the Hartree-Fock method to solve it numerically. $\endgroup$ – iharob Nov 27 '15 at 19:24
  • $\begingroup$ I wonder if there are any mathematical proofs out there which can demonstrate that there are no exact solutions given specific boundary conditions for the He using Schrodinger's or Dirac's equations. $\endgroup$ – dualredlaugh Nov 27 '15 at 19:40
  • $\begingroup$ Maybe three body problems require a different approach in QM. $\endgroup$ – dualredlaugh Nov 27 '15 at 19:41

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