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I Bloch's theorem infinite system?

  1. Apr 26, 2016 #1
    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)
     
  2. jcsd
  3. Apr 28, 2016 #2

    DrDu

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    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
     
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