Energy splitting from spin-orbit interaction.

[soothsayer]

Homework Statement

The spin-orbit interaction splits the hydrogen 4f state into many (a) Identify these states and rank them in order of increasing energy. (b) If a weak external magnetic field were now introduced (weak enough that it does not disturb the spin-orbit coupling), into how many different energies would each of these states be split?

Homework Equations

The Attempt at a Solution

I feel like I understand how to do part b a bit better than part a. My book doesn't seem to be too helpful when dealing with energy splitting due solely to spin-orbit interaction, but goes into much greater detail about the Zeeman effect (presence of weak B field). In part a, I know energy will be split based on whether the angular momenta are aligned or not aligned, but I don't know how to find any split states besides that, and the book implies there are many. Then again, I don't understand why it should be any different than the Zeeman effect, wouldn't the B field perceived by the electron from the proton have the same effect as an external B field? I know the equations I posted give me a size for the difference in energy, but they don't seem to help me with finding the different states.

 

[vela]

What the perturbation term added to the Hamiltonian for the spin-orbit interaction?