How to select the good basis for the special Hamiltonian?

[hokhani]

How to select the good basis for the special Hamiltonian??

For the Hamiltonian H=\frac{P^2}{2\mu} -\frac{Ze^2}{r}+ \frac{\alpha}{r^3} L.S (which we can use L.S=\frac{1}{2} (J^2-L^2-S^2)in the third term) how to realize that the third term,\frac{\alpha}{r^3} L.S, commutes with sum of the first two terms,\frac{P^2}{2\mu} -\frac{Ze^2}{r}, and then conclude that the set of the operators J^2, J_z, L^2, S^2 are the best to work with? (\alpha, Z, e, \mu are constants)

 

[DrClaude]

Usually, you arrive at it by steps. You first have the Hamiltonian in the absence of spin, from which you find that you have quantum numbers $n$, $l$, and $m_l$, the latter usually chosen to be the projection along z. When you add spin, you find that the Hamiltonian is independent of spin, so you describe the eigenstates with
$$
\left| n, l, m_l, s, m_s \right\rangle
$$

When you add spin-orbit coupling, you realize that $m_l$ and $m_s$ are no longer good quantum numbers (conserved quantities), and you try to figure out what is still conserved. the total angular momentum $j$ and its projection $m_j$ are, so you switch to
$$
\left| n, l, s, j, m_j \right\rangle
$$

Note that $j$ and $m_j$ were already good quantum numbers, but since they are redundant with $l$, $m_l$, $s$, and $m_s$ in the absence of spin-orbit coupling, they are not explicitely mentionned.