Proving Bloch's Theorem

by naele
Tags: bloch, proving, theorem
naele is offline
Feb26-12, 02:06 PM
P: 202
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)


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?
Phys.Org News Partner Physics news on
Physicists design quantum switches which can be activated by single photons
'Dressed' laser aimed at clouds may be key to inducing rain, lightning
Higher-order nonlinear optical processes observed using the SACLA X-ray free-electron laser
Colen is offline
May2-12, 03:14 AM
P: 2
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 is offline
May2-12, 04:03 AM
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,101
Quote Quote by naele View Post
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]
Because that is the definition of how the space translation operator acts on a ket.

It may help to write [itex]\theta(r) = H(r) \psi(r)[/itex]. [itex]\theta(r)[/itex] is a ket. What [itex]T(R) \theta(r)[/itex]....

It may help more to consider more traditional function notation for what I believe is being written:
[tex] (T(R) H \psi)(r) = (H \psi)(r + R).[/tex]

Register to reply

Related Discussions
Bloch theorem? Atomic, Solid State, Comp. Physics 2
Bloch's theorem for finite systems ? Atomic, Solid State, Comp. Physics 3
Bloch theorem Advanced Physics Homework 1
counter example to the Bloch's theorem? Quantum Physics 11
periodic potential: Bloch's theorem Quantum Physics 25