# 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?

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#### 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

Staff Emeritus
Gold Member
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:
$$(T(R) H \psi)(r) = (H \psi)(r + R).$$