Proving Bloch's Theorem

[naele]

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)

$$T(R)H(r)\psi(r)=H(r+R)\psi(r+R)=H(r)T(R)\psi(r)$$

I suspect this might be a dumb question, but what allows us to write $T(R)H(r)\psi(r)=H(r+R)\psi(r+R)$, that is why is the translation operator acting on both the Hamiltonian and the wave, and not just on the Hamiltonian?

 

[Colen]

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

 

[Hurkyl]

I suspect this might be a dumb question, but what allows us to write $T(R)H(r)\psi(r)=H(r+R)\psi(r+R)$

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

It may help to write $\theta(r) = H(r) \psi(r)$. $\theta(r)$ is a ket. What $T(R) \theta(r)$...It may help more to consider more traditional function notation for what I believe is being written:
$$(T(R) H \psi)(r) = (H \psi)(r + R).$$