That is genius. Thanks for the link, it makes a lot of sense to me (or at least I think it does!).
To summarise the argument as I see it, it's essentially saying that since no potential barrier can really be infinite the wavefunction of each electron must overlap into other possible potential wells of other atoms. So if you simplify the model and have two electrons in their respective wells, separated by a large potential barrier in the middle, with infinite potential at either end, the wavefunctions of each electron will overlap into the others well. Thus you have to think of the overall wavefunction as a combination of all possible wavefunctions.
Mathematically, it's shown that, when looking at the possible solutions for an individual electron, the wavefunction can have either odd or even parity. When this is combined with the large wavefunction of the electron in the other well, this splits the energies, creating a degeneracy. The degeneracy is only tiny though, so both electrons are seen at being almost exactly the same energy in their respective potential wells. If you were to change the energy level of one of the electrons though, we're forced to conclude that the overlap of the wavefunction into the other potential well would change and consequently the wavefunction of the system as a whole would change.
Spooky action at a distance indeed.