We can rewrite the above transition amplitude using the Wigner-Eckart theorem. Our first step is to rewrite the one-body operator in terms of reduced matrix elements. Since the operator is a spherical tensor we need that the annihilation operator is rewritten as (where \( q \) represents \( j_q \), \( m_q \) etc) $$ \tilde{a}_q=\left(-1\right)^{j_q-m_q}a_{j_q,m_q}. $$ The operator $$ O_{\mu}^{\lambda} = \sum_{pq} \langle p \vert O_{\mu}^{\lambda} \vert q \rangle a^{\dagger}_pa_q, $$ is rewritten using the Wigner-Eckart theorem as $$ O_{\mu}^{\lambda} = \sum_{pq} \langle p \vert \vert O^{\lambda}\vert \vert q \rangle (-1)^{j_p-m_p}\left(\begin{array}{ccc} j_p & \lambda & j_q \\ -m_p & \mu & m_q\end{array}\right)a^{\dagger}_pa_q. $$