Our single-particle wave functions, if we use the harmonic oscillator, do however not contain the quantum numbers \( j \) and \( m_j \). Normally what we have is an eigenfunction for the one-body problem defined as $$ \phi_{nlm_lsm_s}(r,\theta,\phi)=R_{nl}(r)Y_{lm_l}(\theta,\phi)\xi_{sm_s}, $$ where we have used spherical coordinates (with a spherically symmetric potential) and the spherical harmonics $$ Y_{lm_l}(\theta,\phi)=P(\theta)F(\phi)=\sqrt{\frac{(2l+1)(l-m_l)!}{4\pi (l+m_l)!}} P_l^{m_l}(cos(\theta))\exp{(im_l\phi)}, $$ with \( P_l^{m_l} \) being the so-called associated Legendre polynomials.