## Total Angular Momentum and The Spin Orbit Interaction

The spin-orbit interaction (between magnetic dipoles) will play a role in the fine structure of Hydrogen as well as in other problems. It is a good example of the need for states of total angular momentum. The additional term in the Hamiltonian is

If we define the total angular momentum in the obvious way we can write in terms of quantum numbers.

Since our eigenstates of and are also eigenstates of and (but not or ), these are ideal for computing the spin orbit interaction. In fact, they are going to be the true energy eigenstates, as rotational symmetry tells us they must.

Jim Branson 2013-04-22