How many selection rules are there in atomic physics?

Question

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!

Answer 1

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.

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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".