Undergrad Bloch's theorem infinite system?

[thegirl]

Hi,

Does anyone know why k has to be real in an infinite system for bloch's theorem. I understand that the wavefunction becomes unphysical in an infinite system as it diverges. Why does that mean k has to be real?

f(x)=u(x)exp(ikx)

 

[DrDu]

Infinite systems are always quite difficult to discuss. A sound way to discuss this problem is to look at the reduced hamiltonian $H=p^2/2m +V$ acting on the functions u(x) defined on the range [0, a] where a is the length of the elementary cell. It turns out that this hamiltonian is only self-adjoint for periodic boundary conditions so that $u(a)=u(0)\exp(ika)$ where k labels all possible self-adjoint extensions. This is discussed in a pedagogical way in this article:
http://scitation.aip.org/content/aapt/journal/ajp/69/3/10.1119/1.1328351