# Deriving the spin-orbit interaction

#### QMconfused

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
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