Exploring Selection Rules for Hydrogen Atom

[dsdsuster]

I have a couple of questions about selection rules for the hydrogen atom.

When we talk about these rules in an undergraduate context we are usually assuming LS coupling where we ignore spin orbit coupling so orbital and spin angular momentum are essentially independent. Is that correct?

Subsuming the basic l,m_l rules in this case we have
$$\Delta J=\pm 1, \text{except J=0} \rightarrow J=0. \\ \Delta m_j=\pm 1,0 \\ \Delta S=0 \text{ since the electron is always spin half} \\ \text{but } \Delta m_s=\pm 1\text{ since we can transition from spin up to spin down?}$$
If we can indeed transition from spin up to spin down couldn't we possibly have $$\Delta m_j=\pm -2$$ from (m_l=1, up) to (m_l=0,down) then in violation of the above rule?

 

[azntoon]

Yes, that is possible in LS coupling. However, the selection rules dictate that the total angular momentum should change by $\Delta J=\pm 1$, so the only way for this transition to be allowed is if $\Delta m_j=0$ or $\pm 2$, with $\Delta m_l=\pm 1$ and $\Delta m_s=\pm 1$.