This partial wave basis is useful for defining the operator for the nucleon-nucleon interaction, which is symmetric with respect to rotations, parity and isospin transformations. These symmetries imply that the interaction is diagonal with respect to the quantum numbers of total relative angular momentum \( {\cal J} \), spin \( S \) and isospin \( T \) (we skip isospin for the moment). Using the above plane wave expansion, and coupling to final \( {\cal J} \) and \( S \) and \( T \) we get $$ \langle\mathbf{k'}\vert V \vert\mathbf{k}\rangle= (4\pi)^2 \sum_{STll'm_lm_{l'}{\cal J}}\imath^{l+l'} Y_{lm}^{*}(\mathbf{\hat{k}}) Y_{l'm'}(\mathbf{\hat{k}'}) $$ $$ \langle lm_lSm_S|{\cal J}M\rangle \langle l'm_{l'}Sm_S|{\cal J}M\rangle\langle k'l'S{\cal J}M\vert V \vert klS{\cal J}M\rangle, $$ where we have defined $$ \langle k'l'S{\cal J}M\vert V \vert klS{\cal J}M\rangle=\int j_{l'}(k'r')\langle l'S{\cal J}M\vert V(r',r)\vert lS{\cal J}M\rangle j_l(kr) {r'}^2 dr' r^2 dr. $$ We have omitted the momentum of the center-of-mass motion \( \mathbf{K} \) and the corresponding orbital momentum \( L \), since the interaction is diagonal in these variables.