Checking the wikipedia article on band structure, I got caught in major doubts...
They try to give an intuitive explanation of the band structure relying heavily on Pauli Exclusion Principle:
if a large number N of identical atoms come together to form a solid, such as a crystal lattice, the atoms' atomic orbitals overlap. Since the Pauli exclusion principle dictates that no two electrons in the solid have the same quantum numbers, each atomic orbital splits into N discrete molecular orbitals, each with a different energy. Since the number of atoms in a macroscopic piece of solid is a very large number (N~1022) the number of orbitals is very large and thus they are very closely spaced in energy (of the order of 10−22 eV). The energy of adjacent levels is so close together that they can be considered as a continuum, an energy band.
I... am confused. For me, the Pauli exclusion principle states the Following:
2 identical fermions in the same physical system can not be in the same state at the same time
The state of a particle is described by its wave function. Therefore, I don't see how the Pauli principle could Apply to the orbitals of two atoms "far away" in the Crystal...
I mean, to illustrate, if we imagine a 1D monoatomic crystal and take a string of 3 atoms that we label A, B, and C: $$A-B-C$$
I could more or less imagine that the overlap of the valence orbitals of A and B could imply that the Pauli exclusion comes into play for the valence electrons in the A-B system (although I am already not sure this is the right way to describe this).
However, why would the Pauli principle say anything about the electrons of A and C? I mean, the wavefunctions of the electrons of A and C do not overlap. Being "distant from each other", the spatial part of $\psi$ is distinct for those electrons, and therefore the states are already different. No need to call on the Pauli principle, which in this case, should provide no information. No?
Why would the Pauli principle cause a never-ending splitting as we add more and more atoms to the crystal whose electrons have... nothing to do with each other as the distance increases? (ignoring the fact that they could eventually be delocalized as conduction electrons in a metallic crystal)
The way to introduce band structure that I have studied does not make use of the Pauli Principle (in fact, in the book I am thinking of, a chapter on the band structure is placed before the one tackling identical particles). Actually, we could establish the band structure in a single-electron approximation, which suggests that the exclusion principle would have Nothing to do with this result. It only relies on the translational symmetry of the Crystal, deducing the Bloch states and injecting them in the SE to show the bands $E_n(k)$ emerging as a solution.
I understand that the exclusion principle will be important to describe how those bands get filled, but it should not be necessary, in my understanding, to explain that they exist.
I find it highly surprising to find an explanation based on a radically different principle, and I have a hard time imagining that both ideas are equivalent.
Is this Pauli principle approach really correct? If yes, what did I misunderstand?
If yes, can we show that it is equivalent to the well-known periodical-Hamiltonian demonstration?