PX262 - G7c - weak-field zeeman effect

investigating the regime where both spin-orbit coupling and zeeman effect are important

{\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})}}

in calculations, $(\hat{\vec{L}} + 2 \hat{\vec{S}})$ and $(\hat{\vec{L}} + \hat{\vec{S}})$ are needed, which do not align
semi-classically, for the interaction energy:

δ E = B μ_{j z}

where, $μ_{j z}$ is the $z$ -component of the total magnetic moment, ${\vec{μ}}_{j}$

\begin{matrix} (i) & \vec{j} = \vec{l} + \vec{s} \\ (ii) & \vec{j} \cdot \vec{s} = \frac{1}{2} (j^{2} - l^{2} - s^{2}) \end{matrix}

\begin{aligned} {\vec{μ}}_{j} & = \frac{\vec{j}}{j} \cdot \vec{μ} \\ = \frac{e}{2 m j} \vec{j} \cdot (\vec{l} + 2 \vec{s}) \\ = \frac{e}{2 m j} (j^{2} + \vec{j} \cdot \vec{s}) [∵ (i)] \\ = \frac{e j}{2 m} (1 + \frac{\vec{l} \cdot \vec{s} + s^{2}}{j^{2}}) \\ = \frac{e j}{2 m} (1 + \frac{j^{2} - l^{2} - s^{2}}{2 j^{2}}) [∵ (i i)] \end{aligned}

therefore, the interaction energy:

δ E = \frac{e B j_{z}}{2 m} (1 + \frac{j^{2} - l^{2} - s^{2}}{2 j^{2}})

replacing the classical expressions for the magnitudes of the angular momenta by the eigenvalues of the correponding operators:

δ E = g μ_{B} m_{j} B

where,
$μ_{B} = \frac{e ℏ}{2 m}$ is the bohr magneton
$m_{j}$ is the $z$ -component of the total angular momentum
$g = 1 + \frac{j (j + 1) - l (l + 1) + s (s + 1)}{2 j (j + 1)}$ is known as the landé g-factor

if $s = 0$ , $j = l$ , $g = 1$ , which corresponds to all the angular momentum being orbital
conversely, if $l = 0$ , $j = s$ , $g = 2$ , which corresponds to no orbital angular momentum