Saturday, April 2, 2011

How to handle two electrons at once

Yesterday I sketched out the form of the wave function for two electrons in a potential well. It's over a year since I started this blog and I've actually solved a problem or two in that time. It's just hard to believe that I've never really worked out the correct form for this basic problem until now. It's a pretty significant problem for me, because it generalized to a whole list of other problems. Where do we begin? A very good problem to look at is the case of two isolated hydrogen atoms. We can solve each of these individually as single-electron problems, but we really ought to be able to get the same solution by treating it as one big two-electron problem. In fact, I got myself in a lot of trouble last year when I tried to do this and kept coming up with a form of "mini-helium" as a solution. You can go back to my blogs from last winter to see how I got out of that mess. Anyhow, let's begin with the solution we got for two electrons in a box and see how it would be applied to two hydrogen atoms. We can take the diagram from the last blog and just change shape of the function to come up with something like this: This would appear to be the correct representation for the ground state of two hydrogen atoms, taken together. It is interesting that we're not allowed to say that "electron A is here, and electron B is there". We have to allow that each electron can somehow be either here or there. Another interesting aspect is that the electrons must have opposite spin. This is similar to what happens in the ground state of the helium atom. It seems strange to require that the spins be opposite for two separate hydrogen atoms, and in fact we will find that we can get out of this difficulty rather easily. It will turn out that there are three more nearly degenerate states which the electrons can fill, and this gives us the flexibility to specify their spins independently. The most puzzling aspect of this representation, however, turns out to be that the electrons are in a spin singlet state. It's not just that the spin at A is opposite the spin at B. It's that the spin everywhere is identically zero. This is baffling because it's not something we are able to get if we just write down the wave equations of two separate atoms. It's as though by taking them as a complete system we expose new, unexpected behavior.

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