Composite Systems - QM Homework: Equations & Solution

[bon]

Homework Statement

Please see attached problem

Homework Equations

The Attempt at a Solution

Ok so I am 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 H_A commutes with H_int. 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.