Understanding Spin-Orbit Coupling in Hamiltonian

Tianyou
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Hello,
I see a Hamiltonian of spin-orbit coupling, it is like this
H=\frac{1}{2m} \sum_{\alpha} \left( \left( -\partial_{\alpha}^2 - 2i\kappa_{\alpha} \sigma_{\alpha} \partial_{\alpha} \right) + \kappa_{\alpha}^2\right)

Here \hbar = 1. \kappa_{\alpha} is the strength of spin-orbit coupling, and \sigma_{\alpha} (\alpha = x, y, z) are Pauli matrices.
Can someone explain to me where this hamiltonian comes from? What is the meaning of each term?
Thank you.
 
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Where did you see it?
 
I saw this hamiltonian in an article in arXiv titled "Enhancement of condensate depletion due to spin-orbit coupling".
 
Next time, it would be kind of you to give either a link or at least a full reference for an article.
The hamiltonian in that article does not coincide completely with the one of your first post.
However, I don't know where it comes from. Did you check the articles they are citing?
To me, it looks rather dubious. Spin 1/2 for bosons? Certainly not as general as they claim it to be.
 
I am confused for the Spin 1/2 for the bosons before. I checked it and found that in this system there is a pseudo Spin 1/2 states for the bosons.
 
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