Deriving the spin-orbit interaction

Oct 2016
1
0
Hey all,

I'm a first year in engineering grad school who is taking a solid state physics course, and I'm having a tough time with a problem on my first problem set (#3, attached).

I've spoken with my professor, and I was given the following advice:
"write the two wave equations for the two-vectors phi and chi, solve for chi in terms of phi, take the non-relativistic limit. Eigenvalue eqn: H (phi) = E (phi), and you should be able to identify the spin-orbit term in H.

Could anyone explain how to even begin this problem? I've attached the relevant information from my class notes, but I don't have much exposure to these types of manipulations so they are pretty obscure to me.

Thank you!
 

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topsquark

Forum Staff
Apr 2008
3,109
660
On the dance floor, baby!
Hey all,

I'm a first year in engineering grad school who is taking a solid state physics course, and I'm having a tough time with a problem on my first problem set (#3, attached).

I've spoken with my professor, and I was given the following advice:
"write the two wave equations for the two-vectors phi and chi, solve for chi in terms of phi, take the non-relativistic limit. Eigenvalue eqn: H (phi) = E (phi), and you should be able to identify the spin-orbit term in H.

Could anyone explain how to even begin this problem? I've attached the relevant information from my class notes, but I don't have much exposure to these types of manipulations so they are pretty obscure to me.

Thank you!
Interesting, I've never seen this one before.

You have \(\displaystyle \psi = \left ( \begin{matrix} \chi \\ \phi \end{matrix} \right )\).

Suggestions/Outline of hint:
You have the wave equation and you have the two "2-vectors" \(\displaystyle \phi\) and \(\displaystyle \chi\). Insert these into the wave equation and you will have two equations in terms of \(\displaystyle \phi\) and \(\displaystyle \chi\).
\(\displaystyle p_0 \chi - \left ( \sum_{k = 1}^3 p_n \sigma _n \right ) \phi - mc \chi = 0\), for one.

Solve one of these equations for \(\displaystyle \phi\) (for example). Then my question to you is the last part of the hint: How do you take the non-relativistic limit? (Hint: Which is bigger, \(\displaystyle p_0\) or \(\displaystyle \textbf{p}\)?)

-Dan