How can we get a function in terms of \( j \) and \( m_j \)? Define now $$ \phi_{nlm_lsm_s}(r,\theta,\phi)=R_{nl}(r)Y_{lm_l}(\theta,\phi)\xi_{sm_s}, $$ and $$ \psi_{njm_j;lm_lsm_s}(r,\theta,\phi), $$ as the state with quantum numbers \( jm_j \). Operating with $$ \hat{j}^2=(\hat{l}+\hat{s})^2=\hat{l}^2+\hat{s}^2+2\hat{l}_z\hat{s}_z+\hat{l}_+\hat{s}_{-}+\hat{l}_{-}\hat{s}_{+}, $$ on the latter state we will obtain admixtures from possible \( \phi_{nlm_lsm_s}(r,\theta,\phi) \) states.