Seems like people are interested in learning more about how Fermi statistics effect the electron-electron interaction and why we minimize the Helmholtz energy, F= E - TS, to find the equilibrium state of a system coupled to a heat bath (that is, at a well-defined temperature T).
For the latter issue, if someone(s) want to look into that and report back, I think people would be interested. I think that it is possible to connect that directly to the 2nd law or thermo, the one about entropy increasing. The Helmholtz energy plays a big role in a lot of physics so it is worth understanding.
For the e-e interaction issue, perhaps the following calculation might be helpful. Consider a square well from x=0 to 1 nm. Suppose there are two electrons in the wells and that the occupied states are sin(10 pi x) and sin(11 pi x), for example. I think that you can make a symmetric combination of those or an anti-symmetric combination. For those two cases, calculate the expectation value of \(|x_1-x_2|\). See of they are substantially different? Perhaps this will tell us if there are correlations built into these states and if electrons avoid each other more in one than the other? Does this make sense? Feel free to ask questions about it. Here is a normalization integral to get things started. (I could not get W-A to do the integral when the \(|x_1-x_2| = |x-y| \) term was added.)
Integrate [2 sin^2(9 pi x) 2 sin^2(8 pi y)] from x=0 to 1, y=0 to 1
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