Composite Systems - QM

[bon]

1. The problem statement, all variables and given/known data

Please see attached problem


2. Relevant equations



3. The attempt at a solution

Ok so im just stuck on the bit that asks us to write down the appropriate form of the interaction hamiltonian H int for two oscillating particles connected by a weak spring. Is it 1/2 k (xa - xb) ^2?

It asks whether HA commutes with Hint. Not sure about this>? What would be the physical significance?

[graphene]

H_A contains a derivative. H_int contains a polynomial.
Do you think they can commute? You can work it out.

If H_A and H_int commute, and that the total hamiltonian is their sum, you can find common eigenstates between H_A and H_int.
Else, no, you'll have to solve the complete hamiltonian.

[bon]

Is my expression for H_int correct?

I guess they don't commute then?

[graphene]

yes.

yes.