Composite Systems - QM Homework: Equations & Solution

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Homework Help Overview

The discussion revolves around a quantum mechanics problem involving two oscillating particles connected by a weak spring. Participants are examining the appropriate form of the interaction Hamiltonian and its relationship with another Hamiltonian.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are exploring the form of the interaction Hamiltonian and questioning its correctness. There is also discussion about the commutation of two Hamiltonians and the implications of commuting versus non-commuting operators.

Discussion Status

Some participants have provided guidance on the implications of commutation and the potential need to solve the complete Hamiltonian if the two do not commute. There is an ongoing exploration of the correctness of the proposed interaction Hamiltonian.

Contextual Notes

Participants are working under the constraints of a homework assignment, which may limit the information available for discussion. The significance of the commutation relation is also under examination.

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

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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.
 
Is my expression for H_int correct?

I guess they don't commute then?
 
yes.

yes.
 

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