So I have a large confusion with QM as applied to solid state. The following is a summary of what I know, what I think I know, and what I know I don't know. I hope to stir a discussion that will help users of QM, but non-experts, like myself in the future.
Bloch's theorem is statement of symmetry if you're in a perfect lattice (infinite, no defects, zero K). Due to the nature of this symmetry, the wave-function has to have a periodic nature (the exp(ik) part). This is fine, and largely unsurprising (although very elegant).
Here we can talk about $k$ space, which are just the harmonics of the crystal lattice. We can talk about how many electrons/holes they can contain, ect. We call $k$-space "momentum" space because the Fourier transform of position operator is the momentum operator. Physically, $k$ space are modes of lattice vibrations. These modes of the crystal lattice are called phonons.
But the crystal is not infinite, the temperature is not 0, and the lattice is not defect free.
Let' say I have a nanoparticle, of a few thousand atoms more or less placed according to a lattice (which doesn't have to necessarily be the bulk lattice, i.e. it could be a quasi-crystal's lattice). There will be "defects", surface strain, surface energy, possibly a grain boundary, and the shape of the particle will be in constant evolution (non-zero $T$). All of these are "imperfections" that violate the assumptions of Bloch's theorem.
Now Bloch's theorem does not apply, but somehow we want to use it anyway because there is some periodicity within the cluster, at least locally. Simultaneously, we'd like to borrow from our analytic answer to the H atom (s, p, d, f orbitals).
And here I stop understanding anything at all. There is a new concept introduced which is partial/projected density of state (pdos). But what is pdos representing, ie what is the space? What do they mean by "partial" what are they projecting when they speak of "projected"? Why and how do they identify f, d, ect orbitals? How are these charts read?
All this and we haven't begun to speak of spin!