PX262 - G6a - interactions of moments

from here, an electron with an orbital momentum, $\vec{l}$ has a magnetic moment:

μ_{l} = - \frac{e}{2 m_{e}} \vec{l}

the magnetic moment associated with the spin is found to be almost exactly twice using relativistic quantum theory/experiments:

μ_{s} = - \frac{e}{m_{e}} \vec{s}

these moments separately interact with an applied magnetic field, the zeeman effect
they also interact with each other, spin-orbit coupling
combining the hamiltonians of these two interactions, the interaction energy is given by the hamiltonian:

\begin{matrix} {\hat{H}}^{'} = \underset{LS coupling}{\underset{⏟}{f (r) ({\hat{J}}^{2} - {\hat{L}}^{2} - {\hat{S}}^{2})}} + \underset{zeeman effect}{\underset{⏟}{\frac{e}{2 m} B_{0} ({\hat{L}}_{z} + 2 {\hat{S}}_{z})}} \\ where, f (r) = \frac{μ_{0} Z e^{2}}{4 π m^{2} r^{3}} \end{matrix}