Summing up, for for the single-particle case, we have the following eigenfunctions $$ \psi_{njm_j;ls}=\sum_{m_lm_s}\langle lm_lsm_s|jm_j\rangle\phi_{nlm_lsm_s}, $$ where the coefficients \( \langle lm_lsm_s|jm_j\rangle \) are the so-called Clebsch-Gordan coeffficients. The relevant quantum numbers are \( n \) (related to the principal quantum number and the number of nodes of the wave function) and $$ \hat{j}^2\psi_{njm_j;ls}=\hbar^2j(j+1)\psi_{njm_j;ls}, $$ $$ \hat{j}_z\psi_{njm_j;ls}=\hbar m_j\psi_{njm_j;ls}, $$ $$ \hat{l}^2\psi_{njm_j;ls}=\hbar^2l(l+1)\psi_{njm_j;ls}, $$ $$ \hat{s}^2\psi_{njm_j;ls}=\hbar^2s(s+1)\psi_{njm_j;ls}, $$ but \( s_z \) and \( l_z \) do not result in good quantum numbers in a basis where we use the angular momentum \( j \).