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.

 

[paranormal]

Hello there,

Thank you for reaching out for help with this problem. It looks like you are working on a composite system in quantum mechanics, which can be a complex topic. Let's break down the problem and see if we can understand it better.

First, the problem asks you to write down the appropriate form of the interaction Hamiltonian, denoted as Hint. This Hamiltonian describes the interaction between two oscillating particles connected by a weak spring. The equation you have written, 1/2 k (xa - xb)^2, is the correct form for this interaction Hamiltonian. This represents the potential energy of the spring, where k is the spring constant and xa and xb are the position operators for the two particles.

Next, the problem asks whether the Hamiltonian for the entire system, denoted as HA, commutes with Hint. This means whether the two operators can be measured simultaneously without affecting each other's measurements. In this case, the physical significance is that the two particles are not interacting strongly enough for their measurements to affect each other. This is an important concept in quantum mechanics and can have implications for how we understand and describe complex systems.

I hope this helps clarify the problem for you. If you have any further questions or need more assistance, please don't hesitate to ask. Remember, in science, it's important to take the time to fully understand a problem before moving on to the solution. Keep up the good work!