- #1
naele
- 202
- 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?
[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?