How does Bloch's theorem generalize to a finite sized crystal?

Question

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 1

You may think this way: take a perfect infinite crystal where Bloch theorem perfectly work and add potential which makes real crystal finite. Next question you may ask how this potential is "seen" by quasiparticles which have been obtained from infinite crystal consideration. This procedure is perfectly self-consistent and is applicable in all cases.

Also, using this approach you may get some idea when Bloch theorem stops working. However, there is no definite answer to this question: it depends on which value you are interested in. If you are interested in momentum conservation, fourier spectrum of this confining potential should be small at wavevectors you consider. If you consider energy, levels spacing due to quantum confinement should be small compared to some characteristic energy, etc.

Answer 2

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 $1/L$.

Answer 3

For one dimensional crystal, when we have periodic boundary condition, we may think that the atoms are arranged in a circle, and so each lattice point will be equal to other lattice point and the translation symmetry is conserved.

And for three dimensional crystal, due to periodic boundary condition, each atom will feel the same in the lattice. They are in a finite crystal and then equal to the atom one crystal away from the atom. So the translation symmetry by Bravais lattice is still conserved. So the Bloch theorem still applies here.