Similarly, using $$ \left(\begin{array}{ccc} J & 1 & J \\ -M & 0 & M'\end{array}\right)=(-1)^{J-M}\frac{M}{\sqrt{(2J+1)(J+1)}}\delta_{MM'}, $$ we have that $$ \langle \Phi^J||{\bf J}||\Phi^{J}\rangle=\sum_{M,M'}(-1)^{J-M}\left(\begin{array}{ccc} J & 1 & J' \\ -M & 0 & M'\end{array}\right)\langle \Phi^J_M|j_Z|\Phi^{J'}_{M'}\rangle=\sqrt{J(J+1)(2J+1)} $$ With the Pauli spin matrices \( \sigma \) and a state with \( J=1/2 \), the reduced matrix element $$ \langle \frac{1}{2}||{\bf \sigma}||\frac{1}{2}\rangle=\sqrt{6}. $$ Before we proceed with further examples, we need some other properties of the Wigner-Eckart theorem plus some additional angular momenta relations.