Understanding Spin-Orbit Coupling in Hamiltonian

In summary, the conversation discusses a Hamiltonian of spin-orbit coupling, which is used in the article "Enhancement of condensate depletion due to spin-orbit coupling" on arXiv. The Hamiltonian is represented by H=\frac{1}{2m} \sum_{\alpha} \left( \left( -\partial_{\alpha}^2 - 2i\kappa_{\alpha} \sigma_{\alpha} \partial_{\alpha} \right) + \kappa_{\alpha}^2\right) and includes parameters such as \kappa_{\alpha} and \sigma_{\alpha}. The meaning of each term is unclear and the conversation suggests checking the cited
  • #1
Tianyou
4
0
Hello,
I see a Hamiltonian of spin-orbit coupling, it is like this
H=[itex]\frac{1}{2m} \sum_{\alpha} \left( \left( -\partial_{\alpha}^2 - 2i\kappa_{\alpha} \sigma_{\alpha} \partial_{\alpha} \right) + \kappa_{\alpha}^2\right)[/itex]

Here [itex]\hbar = 1. \kappa_{\alpha} [/itex] is the strength of spin-orbit coupling, and [itex] \sigma_{\alpha} (\alpha = x, y, z)[/itex] 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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  • #2
Where did you see it?
 
  • #3
I saw this hamiltonian in an article in arXiv titled "Enhancement of condensate depletion due to spin-orbit coupling".
 
  • #4
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.
 
  • #5
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.
 

1. What is spin-orbit coupling in the context of Hamiltonian?

Spin-orbit coupling is a phenomenon in quantum mechanics where the spin of an electron interacts with its orbital motion around the nucleus. In the context of Hamiltonian, it is the term in the Hamiltonian operator that describes this interaction.

2. How does spin-orbit coupling affect the energy levels of an atom?

Spin-orbit coupling causes the energy levels of an atom to split into two slightly different levels. This is because the spin and orbital angular momentum of an electron interact, resulting in a lifting of the degeneracy of energy levels.

3. What is the physical significance of spin-orbit coupling?

The physical significance of spin-orbit coupling is that it is responsible for various phenomena, such as fine structure in atomic spectra, the Zeeman effect, and the spin Hall effect. It also plays a crucial role in understanding the electronic structure of materials.

4. How is spin-orbit coupling described in the mathematical framework of Hamiltonian?

Spin-orbit coupling is described in Hamiltonian through the spin-orbit operator, which is a vector operator that accounts for the interaction of the electron spin with its orbital motion. This operator is added to the Hamiltonian to account for the spin-orbit coupling energy term.

5. Can spin-orbit coupling be observed in everyday life?

Yes, spin-orbit coupling can be observed in everyday life, especially in materials that have strong spin-orbit interactions, such as heavy elements like lead and gold. It is also responsible for the color of gold, as the spin-orbit coupling between gold atoms causes them to absorb and reflect different wavelengths of light.

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