Can the Born-von Karman boundary conditions be ignored without leading to contradictions? [duplicate]

Question

We use the Bloch theorem and requirements of continuity of the wave functions to get the relation $E \times k$ for the bands in the 1D model of Kronig Penney, that also shows the band gaps.

All this is done without any assumption about which are the individual quantum states. They are defined after the imposition of the Born-von Karman boundary condition, resulting in equally $k$-spaced states. But could they be less dense for small $k$'s and denser at the extremities of the Brillouin zone for example? Or some contradiction arises in this case?

My question is: What empirical information is related to BvK? Why is necessary to postulate that the Bloch waves must have a $Na$ periodicity for the same $N$ and all $k$'s?

Answer 1

Boundary conditions are necessary to provide meaning to states and operators. In particular, in the case of a periodic solid, an alternative to Born-von Karman (BvK) boundary conditions would be working with an infinite lattice from the beginning. However, such a choice would make the corresponding wavefunctions not element of the $L_2({\mathbb R}^3)$ Hilbert space. If we renounce to periodicity, Bloch's theorem canot be used.

𝑁𝑎 periodicity of the wavefunctions for the same 𝑁 and all 𝑘's is required to ensure the self-adjointness of the Hamiltonian (precisely for the same reason continuity of wavefunctions is requested).