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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!

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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".

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  • $\begingroup$ Isn't also $M_J=0 \to M_J=0$ forbidden if $\Delta J=0$ in LS coupling (as stated on page 6 of ps.uci.edu/~cyu/p224/LectureNotes/lecture6/lecture6.pdf and in this Q physics.stackexchange.com/questions/288830/…)? I'm currently trying to figure out why this is so rarely stated, if it's true $\endgroup$
    – llama
    Nov 2, 2023 at 21:11
  • $\begingroup$ @llama That rule is present on the Wikipedia page linked in this answer, which the answer does not in any way pretend to summarize in full. I don't really know what you mean by it being "rarely stated" if it is so easily found on Wikipedia. $\endgroup$ Nov 2, 2023 at 21:22
  • $\begingroup$ Well, I'm the one who added it to that table on the wikipedia page, it wasn't there a year ago. It's just that I received an edit for something I'm working on (from someone who I would expect to know these things) which said this selection rule was not valid, so I had to convince myself that they were wrong, and it was harder to find that confirmation than I expected. Of course I'd convinced myself before I added it to the wiki table, but I couldn't remember how I'd done so. $\endgroup$
    – llama
    Nov 3, 2023 at 15:09
  • $\begingroup$ I'm just saying if you google "electric dipole selection rules" and look at a sample of results, almost all of them don't explicitly state that rule, they leave it implied by the others. I don't understand why they do this, which makes me second-guess all the ways I've convinced myself of it. $\endgroup$
    – llama
    Nov 3, 2023 at 15:21

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