I would be fine with a one dimensional lattice for the purpose of answering this question. I am trying to figure out what more general theorem (if any) gives Bloch's theorem as the number of unit cells-->infinity.
Answer
Bloch's theorem generalizes nicely to a finite size crystal if we take periodic boundary conditions (pbc). If we have pbc than the a translation by one unit cell is still a symmetry of the system and so Bloch's theorem will apply.
The only difference will be that the quasimomentum $q$ will only be allowed to take certain discrete values since the wavefunciton must periodic over the entire system. Specifically $qL = 2\pi$ where $L$ is the length of the system. This is all exactly analogous to taking a free particle and putting it in a periodic box.
Periodic boundary conditions are not physical of course (except for rings of atoms) but as with the particle in a box they give the essential structure. In this case the point is that the quasimomentum is still a good quantum number locally, but the energy spectrum is now discrete with energy spacing like $1/L$.
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