(a) From Eq. (9.15) the Land (g)-factor is [begin{equation*}g_{J}=1+frac{J(J+1)-L(L+1)+S(S+1)}{2 J(J+1)} . tag{9.15}end{equation*}] Show that (g_{J}) can be...

(a) From Eq. (9.15) the Landé \(g\)-factor is

\[\begin{equation*}g_{J}=1+\frac{J(J+1)-L(L+1)+S(S+1)}{2 J(J+1)} . \tag{9.15}\end{equation*}\]

Show that \(g_{J}\) can be written in the equivalent form

\[g_{J}=\frac{3}{2}+\frac{(S-L)(S+L+1)}{2 J(J+1)} .\]

(b) Consider the case \(L=S\). Evaluate the sum of the \(g_{J}\) over the allowed values of \(J\).

(c) Consider the case \(L=3, S=1\). Evaluate the sum of the \(g_{J}\) over the allowed values of \(J\).