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I am having troubles with understanding the allowed k vectors in a crystal.

Bloch's theorem gives us discrete energy bands for each wave vector k.

However, only discrete k vectors are allowed. Using periodic boundary conditions, the discreteness is easy to show.

But I am having a hard time in understanding how the discreteness follows from the condition that the wave amplitude must be zero at the boundaries of a crystal. Is the latter even correct? The electrons are confined to a Lx*Ly*Lz cube so that their amplitude should drop to zero at the boundaries. Can anyone offer a proof in how that translates to discrete k vectors? The volume boundaries are a continuum, so I don't really know how that would work.

Best,

SL