Help with Commutators: H(P,Q) & H(P,Q')

[siyphsc]

Homework Statement

(H - hamiltonian, P - momentum, Q- position)
Given two operators Q and Q', I have shown that [H(P, Q), Q] = [H(P, Q), Q']. I was wondering if this meant that I could assume that an energy spectrum found from H(P, Q) could be related to that of H(P, Q'). I am under the impression that the spectrum would be shifted over by a constant (multiplied by identity).

Homework Equations

Given above.

The Attempt at a Solution

The above problem comes from my attempt to solve a more complicated problem.

 

[NateD]

I am trying to solve a time-independent Schrodinger equation using the Heisenberg picture. I have been able to find that [H(P,Q), Q] = [H(P, Q), Q'] (where O is the Hamiltonian, P is the momentum and Q is the position). This means that the Hamiltonian is invariant under translation, i.e. the energy spectrum of H(P,Q) can be related to that of H(P,Q') by a constant shift, as you stated. However, this does not necessarily mean that the eigenfunctions of H(P,Q) and H(P,Q') are related. In order to determine if they are, you will need to use the eigenvalue equation for the Hamiltonian to calculate the eigenfunctions for each Hamiltonian. If the eigenfunctions are related, then it means that the energy spectrum can be related by a constant shift.