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[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?