Can Periodic Hamiltonian and Translation Operators Share Eigenstates?

[eNtRopY]

Prove that if a periodic Hamiltonian operator commutes with the translation operator then these two operators can have simultaneous eigenstates.


{{ H(x) = H(x+a) }
& { T(a)f(x) = f(x+a) }
& { [T(a),H(x)] = 0 }
}
--> {{[uni][psi] [subset] complex functions, there exists [psi] [subset] complex functions
such that { H(x)[psi](x) = E[psi](x) }
& { T(a)[psi](x) = c(a)[psi](x) }
}

Thanks dudes.

eNtRopY

 

[jonnylane]

is the periodic hamiltonian a harmonic oscillator function?

 

[eNtRopY]

No it's independent of time. I mean that it is periodic in space... like a Bloch function.

V(x+a) = V(x)

Actually, I believe I figured out the answer, but if you guys want to post your answer that's cool too.

eNtRopY

 

[quantumdude]

Originally posted by eNtRopY
Prove that if a periodic Hamiltonian operator commutes with the translation operator then these two operators can have simultaneous eigenstates.

Are you sure that's the problem? I ask because any two operators that commute can have simultaneous eigenstates; the "periodic" condition has nothing to do with it.

Methinks you are supposed to be finding an explicit expression for what exactly those eigenstates are.


In this case, |θ>=Σe^inθ|n> (n goes from -[oo] to +[oo]) is an eigenstate of τ(a) with eigenvalue e^-iθ.

This is from Sakurai, Modern Quantum Mechanics pp. 261-263.