How many selection rules are there in atomic physics? How many selection rules are there in atomic physics and how should they be used? Sometimes selection rules are governed by $\Delta j=0,\pm 1$. Sometimes $\Delta m_l=0,\pm 1$ is used. Sometimes $\Delta l=\pm 1$ is used. I never derived them. Therefore, atomic physics looks like a complete mess. Please help! Thanks!
 A: As it turns out, there is an infinite hierarchy of selection rules for the interaction of atomic systems with electromagnetic radiation - but, in practice, you only use one of them, the dipole selection rule.
Within the dipole selection rule, there are also a few different flavours that depend on the angular-momentum structure of the atom itself - more specifically, whether the orbital and spin angular momenta of the different subshells are combined via LS coupling or via some beyond-LS-coupled scheme like intermediate coupling or even full jj-coupling.
The end result is already well described by Wikipedia but, to explain it in a tiny bit more depth:


*

*Electric dipole transitions are always forced to obey:


*

*$\Delta J=0,\pm 1$, with transitions $0\to0$ on $J$ forbidden, and

*$\Delta M_J=0,\pm 1$.


*If the atomic structure is LS-coupled then, in addition to the above, $\Delta L = \pm 1$.


You should also keep in mind that throughout atomic physics, the usage of lowercase symbols for angular-momentum observables ($l,m_l,s,j$, as used in your question) implies that they are single-electron observables; if more than one electron is involved, then an upper-case letter (so: $L,M_L,S,J$) is required. The selection rules as phrased above are in terms of the latter and not the former.

As for this,

Therefore, atomic physics looks like a complete mess,

I think what you really meant to say was "I have not yet had time to actually read up on a systematic account of atomic physics and, as such, I am not yet able to fully understand the context of the usage of certain terminology, which leaves me confused". Fortunately, the fix is easy: take the time to actually read up on a systematic account of atomic physics before complaining that it "looks like a mess".
