Proving Bloch's Theorem

  1. One of the more common ways of showing that a Hamiltonian with periodic potential commutes with the translation operator is to write the following (like Ashcroft and Mermin p. 133)

    [tex]
    T(R)H(r)\psi(r)=H(r+R)\psi(r+R)=H(r)T(R)\psi(r)
    [/tex]

    I suspect this might be a dumb question, but what allows us to write [itex]T(R)H(r)\psi(r)=H(r+R)\psi(r+R)[/itex], that is why is the translation operator acting on both the Hamiltonian and the wave, and not just on the Hamiltonian?
     
  2. jcsd
  3. I think its because the potential is periodic then the Hamiltonian is too: H(x)=H(x+a), you can then sub this in directly and the translation operator now just acts on psi
     
  4. Hurkyl

    Hurkyl 16,090
    Staff Emeritus
    Science Advisor
    Gold Member

    Because that is the definition of how the space translation operator acts on a ket.

    It may help to write [itex]\theta(r) = H(r) \psi(r)[/itex]. [itex]\theta(r)[/itex] is a ket. What [itex]T(R) \theta(r)[/itex]....


    It may help more to consider more traditional function notation for what I believe is being written:
    [tex] (T(R) H \psi)(r) = (H \psi)(r + R).[/tex]
     
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