I'm going to digress for a moment from the discussion on electric motors to return to the question of the double-potential well, which has surfaced on a couple of discussion groups in the last month. A year ago I described the wave function for the two-electron well...not two wells with two electrons, but one well with two electrons. And then I generalized this to the case of two hydrogen atoms far apart, which is basically the equivalent problem to the double-potential well.
(EDIT: It turns out I'm going to make so many mistakes in what's I'm about to do, that I should probably delete the whole post and start again from scratch. And yet I'm not going to do that. I think it's more realistic to show how the thought process works including the false steps. If you want to skip ahead to the cleaned up version, just click on the next post.)
At that time, I described only the lowest-energy state, which happens to be the spin-singlet state. There are actually three more spin-triplet states that need to be considered. For the two-electron well, these states are significantly higher in energy, but for the case of two hydrogen atoms far apart (the double well) they are so close in energy as to be virtually degenerate. Last year when I sketched them I used a couple of different graphic styles, and this year I've come up with yet another way of drawing the pictures. I hope this one is self-explanatory: (EDIT: This picture is still correct!)
There are three more states needed to complete the picture, and they are the triplet states. The first two are nice and easy. (EDIT: Here's where I start screwing up.) First, both electrons with spin up:
And finally, the state that confuses people: the triplet state with spin zero. What makes this state physically different from the singlet state? It's a funny question. Here is what the state looks like:
If I've drawn these right, and I think I have, (EDIT: I know I haven't! Still working on fixing them...) then anything that two hydrogen atoms can do has to be representable as the superposition of two or more of these systems. For example, if you want to say that this atom has spin up and the other one has spin down, you have to figure out how to generate that outcome as a superposition of these four basis states.
I don't have time to do that right now but I'll come back to it later.
15 MINUTES LATER: Okay, I said the triplet state was confusing, and it was. So confusing that I got it wrong too. Here is the REAL triplet state:
When I come back, once I've got all the basis states fixed up, we'll see what we can do with them. The first problem is to show how we can have one atom with spin up and the other with spin down. That will be easy. What gets harder is to show how we can put the basis states together to form any arbitrary combination of spin states.