Using the Wigner-Eckart theorem $$ \langle \Phi^J_M|O^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle\equiv(-1)^{J-M}\left(\begin{array}{ccc} J & \lambda & J' \\ -M & \mu & M'\end{array}\right)\langle \Phi^J||O^{\lambda}||\Phi^{J'}\rangle, $$ we can then define $$ \langle \Phi^J||O^{\lambda}||\Phi^{J'}\rangle = \lambda^{-1}\sum_{j_pj_q} \langle p \vert \vert O^{\lambda} \vert\vert q \rangle \langle \Phi_{M}^{J}\vert\vert \left[ a_{j_p}^{\dagger}\tilde{a}_{j_q}\right]^{\lambda}\vert \vert \Phi_{M'}^{J'}\rangle. $$ The quantity to the left in the last equation is normally called the transition amplitude or in case of a decay process, simply the decay amplitude. The quantity \( \langle \Phi_{M}^{J}\vert \lambda^{-1}\left[ a_{j_p}^{\dagger}\tilde{a}_{j_q}\right]^{\lambda}_{\mu} \vert \Phi_{M'}^{J'}\rangle \) is called the one-body transition density while the corresponding reduced one is simply called the reduced one-body transition density. The transition densities characterize the many-nucleon properties of the initial and final states. They do not carry information about the transition operator beyond its one-body character. Finally, note that in a shell-model calculation it is actually \( \langle \Phi^J_M|O^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle \) which is calculated.