The monopole term

An important ingredient in studies of effective interactions and their applications to nuclear structure, is the so-called monopole interaction, normally defined in terms of a nucleon-nucleon interaction \( \hat{v} \) $$ \begin{equation} \bar{V}_{\alpha_p\alpha_q} = \frac{\sum_{J}(2J+1) \langle (\alpha_p\alpha_q)J | \hat{v} | (\alpha_p\alpha_q)J \rangle }{\sum_{J}(2J+1)}, \tag{20} \end{equation} $$ where the total angular momentum of a two-body state \( J \) runs over all possible values. In the above equation we have defined a nucleon-nucleon interaction in a so-called angular-momentum coupled representation with the symbol \( \alpha_{p,q} \) representing all possible quantum numbers except the magnetic substates \( m_{j_{p,q}} \) The monopole Hamiltonian can be interpreted as an angle-averaged matrix element. We have assumed that the single-particle angular momenta \( j_p \) and \( j_q \) couple to a total two-particle angular momentum \( J \). The summation over \( J \) with the value \( 2J+1 \) can be replaced by \( \sum_J(2J+1)=(2j_p+1)(2j_q+1) \) if \( \alpha_p\ne \alpha_q \). If \( \alpha_p=\alpha_q \) we can generalize this equation to, assuming that our states can represent either protons or neutrons, $$ \begin{equation} \sum_{J}(2J+1)=(2j_p+1)(2j_q+1-\delta_{\alpha_p\alpha_q}). \tag{21} \end{equation} $$