Uncoupled and coupled bases for electrons in hydrogen atom?

Question

I'm given that for an electron in a hydrogen atom, $L=2$ and $S=1/2$ (quantum numbers associated with $L^2$ and $S^2$). I'm also given that for the uncoupled representation, the basis function is $|L,m_L\rangle |S,m_S\rangle$ and for the coupled, $|J,m_J\rangle$ ($J$ associated with total angular momentum operator $J^2$). I need to find allowed values for $m_L$, $m_S$, $J$ and $m_J$.

My attempted solution: since $S=1/2$, the allowed values of $m_S$ are $m_S=-S,-S+1,\ldots,S=-1/2,1/2$. and since $J=L+M$, $J=5/2$ and $m_J=m_S+m_L$. But I'm not sure how to find $m_L$.

I also saw on an MIT paper that for the uncoupled representation, $[L^2,S^2]=0$. But I'm very shaky with commutator relations so I wasn't sure how to proceed from here. As well I thought I might use Clebsch-Gordan coefficients, but since I don't know $m_L$, I didn't think I could use the table.

Any hints?

Answer 1

When adding angular momentum $S$ and $L$ to get $J$, $J$ can range from absolute value of $|L-S|$ to $L+S$ in integer steps. Then once you have $J$ you can figure the allowed values of $m_j$ just like you do for $S$ and $m_s$