Even though the rotational symmetry is not exact, the approximate degeneracies are sufficient to extract $j$. States with the same orbital angular momentum $\ell$ tend to cluster, as expected from the non-relativistic quark model, even though $\ell$ is not a good quantum number, which is also helpful for identifying states. []{data-label="fig:raw_spectrum"}](Figures/spectrum_a0p60_kap1p39_mb4p90_style2){width="60.00000%"}

The effective Hamiltonian (\[eqn:Heff\]) is diagonalized for various $m_j$ sectors. Fig. \[fig:raw\_spectrum\] shows a representative spectrum as a function of $m_j$. The spectrum is symmetric with respect to $\pm m_j$, a consequence of the mirror parity symmetry (\[eqn:parity\]). The discrete quantum numbers $m_P = (-i)^{2j}\mathsf P$ and $\mathsf C$ are computed to help identify states as mentioned. Total spin $\langle\vec s^2\rangle=s(s+1)$ as an approximate quantum number is also exploited. States with the same $j$ but different $m_j$’s are not exactly degenerate owing to the violation of the rotational symmetry. As is seen in Fig. \[fig:raw\_spectrum\], the approximate degeneracies are easily visible, at least for low-lying states. So the multiplicities, together with $m_P$, $\mathsf C$, $s$ and the constraints: $$|\ell - s| \le j \le \ell+s, \quad \mathsf P=(-1)^{\ell+1}, \quad \mathsf C = (-1)^{\ell+s},$$ can be employed to deduce the full set of quantum numbers $n\,{}^{2s+1}\!\ell_j$ or $j^{\mathsf P\mathsf C}$, 