For an electron in an atom, the angular momentum \( \mathbf{L} \) and spin \( \mathbf{S} \) couple to form a constant total angular momentum \( \mathbf{J} \). Without a spin-orbit interaction the quantum numbers \( m_l \) and \( m_s \) are degenerate. But now the energy levels depend on the relative orientation of \( \mathbf{L} \) and \( \mathbf{S} \), so \( m_l \) and \( m_s \) are no longer good quantum numbers. Instead, the operator \( \mathbf{\hat{J}} \) gives a new quantum number \( j \). Similarly, the operator \( \mathbf{\hat{J}}_z \) gives rise to a new quantum number \( m_j \). Hence, we obtain a set of operators:
\(\mathbf{\hat{S}}^2\mathbf{\Psi}=\hbar^2 s(s+1)\mathbf{\Psi}\)
\(\mathbf{\hat{L}}^2\mathbf{\Psi}=\hbar^2 l(l+1)\mathbf{\Psi}\)
\(\mathbf{\hat{J}}^2\mathbf{\Psi}=\hbar^2 j(j+1)\mathbf{\Psi}\)
\(\mathbf{\hat{S}}_z\mathbf{\Psi}=\hbar m_s\mathbf{\Psi}\)
\(\mathbf{\hat{L}}_z\mathbf{\Psi}=\hbar m_l\mathbf{\Psi}\)
\(\mathbf{\hat{J}}_z\mathbf{\Psi}=\hbar m_j\mathbf{\Psi}\)
\( m_s = -s,\dots ,s \)
\( m_l = -l,\dots ,l \)
\( m_j = -j,\dots ,j \)