I have posted questions similar to this over the past week, but I have never had a satisfactory answer. It is about the fermi distribution. From hyperphysics: "We picture all the levels up to the Fermi energy as filled, but no particle has a greater energy. This is entirely consistent with the Pauli exclusion principle where each quantum state can have one but only one particle." First of all this picture has got to be wrong. In QM there is no such thing as this or that particle, though I guess it is understandable that you picture the fermi distribution as a lot distinct particles occupying the different energystates up to the fermi energy. I however, want to understand the fermi distribution from a QM point of view. It's all got to start with the wave functions antisymmetrization requirement for fermions (after all the pauli principle is derived from this). However, it just seems that all the derivations of the Fermi distribution that I have seen this is not the starting point but rather use the same idea as the quote above. That is: Instead of starting from the wave functions they will say something like this: Suppose we have n particles. For each energy eigenstate only one electron can occupy it due to the Pauli principle (spin is neglected here). But this is QM! A particle does not necessarily have to be in an energy eigenstate. Let's label the first two energy-eigen states by e1 and e2. Then the wave functions: ae1 + be2, ce1 + de2, fe1+ge2 ..... are all allowed. So how on earth can it be the right approach to simply start with the general idea that each electron can only occupy one eigenstate and work the combinatorics from there (noone would know if God has distributed the electrons among eigenstates or among linear combinations of eigenstates).. This annoys me so much. Please try to explain what I am thinking wrong.
I think the main reason behind this is the mean field approach: You assume that the interaction of the particles can be taken into account via an effective potential which is more or less the same for each particle, especially in the limit of infinite system size. So you expect to find the ground state of the many electron system to be composed of one electron states where the lowest states are filled up consistent with the Pauli principle. You are right that this does not always work. For example in superconductors an infinitely weak attractive attraction of the electrons is enough to produce a ground state which can be shown to be orthogonal to the mean field ground state. That's why it took so long to find a microscopic explanation for superconductivity and BCS getting the nobel prize.