= Solution
{c}
Within a fixed $L,S,J$ <angular momentum> multiplet, the component of $\mathbf L$ along $\mathbf J$ follows from $2\mathbf L\cdot\mathbf J=\mathbf J^2+\mathbf L^2-\mathbf S^2$. The corresponding result for $\mathbf S$ follows by interchanging $L$ and $S$. For the <quantum state> $|LSJM_J\rangle$,
$$
\begin{aligned}
\langle L_z\rangle&=M_J\frac{J(J+1)+L(L+1)-S(S+1)}{2J(J+1)},\\
\langle S_z\rangle&=M_J\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}.
\end{aligned}
$$
These are dimensionless <expectation values> in the convention of part (a). Substituting into the first-order <time-independent perturbation theory> formula $\Delta E=\langle H_Z\rangle$ gives
$$
\boxed{\Delta E=g_J\mu_BBM_J,\qquad
g_J=1+(g_s-1)\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}.}
$$
For $g_s=2$, the <Landé g-factor> becomes
$$
\boxed{g_J=1+\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}.}
$$
The <magnetic sublevels> must first be diagonalized within a degenerate multiplet; choosing $M_J$ along the <magnetic field> already does this. For $J=0$ the displayed fraction is undefined, but all vector <expectation values> vanish and there is no first-order <Zeeman splitting>. A higher-order shift may remain.
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