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)(adsbygoogle = window.adsbygoogle || []).push({});

[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?

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# Proving Bloch's Theorem

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