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
are also eigenstates of
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.