Uncertain Hamiltonian and Values Therein

[Kreizhn]

Hey all,

I've got a Hamiltonian of the form

$$H = \omega (\sigma_z^1 - \sigma_z^2) + J \sigma_z^1 \sigma_z^2$$

where $\omega$ is a frequency ( I think), J is the indirect dipole-dipole coupling, and $\sigma_z^i$ is the Pauli Z operator on the ith particle.

Does anybody know what this Hamiltonian represents? Where it's used? Or values for $\omega, J$? I can look up J coupling easily enough given that I randomly choose two interacting particles/molecules, but I'm really uncertain as to what $\omega$ represents in this case and what values it should take.

 

[peteratcam]

How about a two site Ising model in a staggered magnetic field?
Or a million other possible interpretations - it looks like a toy model.

 

[Kreizhn]

From what I understand it's being implemented physically so I don't think it's too "idealized". Are there omega values that go along with that by chance?

 

[peteratcam]

I don't know, your question is very odd. *If* that Hamiltonian is realized somewhere, then the realization will have definite values for the parameters. But you can't point to a Hamiltonian and demand to know the values of the parameters without specifying a particular system. In any case, it's only got 4 states, and doesn't even need to be diagonalized: I don't understand your interest.

 

[Kreizhn]

I don't believe I made any such demands. I'm just inquiring as to if somebody would recognize the Hamiltonian and hence could point out plausible values of omega. It's not even that I am looking for concrete values, but more the physical significance of the omega term from which I can find the values myself. Sadly, I am more of a mathematician than a physicist and so sometimes I require someone to point me in the right "physical" direction.

Also, I fail to see how there are only four states. There may be four basis states, but that is certainly not an exhaustive list. This is, in fact, not the entire Hamiltonian but instead representing a subsystem that I do not recognize. I also don't recall saying I was interested in what states become of this Hamiltonian.

I mean no offense when I say that your scope as to the possible applications of such a Hamiltonian and why it might be interesting are quite narrow. If you would like the full story I would be more than happy to explain, though I do not believe it will be enlightening.

 

[peteratcam]

I'll explain where I'm coming from as a physicist - normally one starts with a physical system in mind, then writes down a plausible model Hamiltonian to work from. But here it is the opposite, you have presented a Hamiltonian, and want to know a plausible physical system.

The Hamiltonian you have written is already diagonal in the obvious choice of basis, and there are 4 eigenstates.

This is, in fact, not the entire Hamiltonian but instead representing a subsystem that I do not recognize.

Does it commute with the rest of the Hamiltonian?

I mean no offense when I say that your scope as to the possible applications of such a Hamiltonian and why it might be interesting are quite narrow. If you would like the full story I would be more than happy to explain, though I do not believe it will be enlightening.

If you explain further then there is certainly more chance that someone here will be able to help you, because at the moment, I cannot make sense of your question.