The Wigner-Eckart theorem states that the expectation value for an irreducible spherical tensor can be written as $$ \langle \Phi^J_M|T^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle\equiv\langle \lambda \mu J'M'|JM\rangle\langle \Phi^J||T^{\lambda}||\Phi^{J'}\rangle. $$ Since the Clebsch-Gordan coefficients themselves are easy to evaluate, the interesting quantity is the reduced matrix element. Note also that the Clebsch-Gordan coefficients limit via the triangular relation among \( \lambda \), \( J \) and \( J' \) the possible non-zero values.

From the theorem we see also that $$ \langle \Phi^J_M|T^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle=\frac{\langle \lambda \mu J'M'|JM\rangle\langle }{\langle \lambda \mu_0 J'M'_0|JM_0\rangle\langle }\langle \Phi^J_{M_0}|T^{\lambda}_{\mu_0}|\Phi^{J'}_{M'_0}\rangle, $$ meaning that if we know the matrix elements for say some \( \mu=\mu_0 \), \( M'=M'_0 \) and \( M=M_0 \) we can calculate all other.