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The selection rules for hydrogen are: $\Delta l=\pm1$ and $\Delta m=0,\pm1$. The first makes intuitive sense because of the conservation of angular momentum and the fact that a photon has spin 1. But does there exist an intuitive explanation for the $m$ selection rule as well?

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  • $\begingroup$ I think that your intuitive argument might be misleading as a photon can carry higher values of orbital angular momentum: en.wikipedia.org/wiki/Orbital_angular_momentum_of_light $\endgroup$
    – Mauricio
    Feb 3, 2022 at 18:06
  • $\begingroup$ @Mauricio but as far as I understand photons with different orbital angular momentum can only created artificially which should be here no problem. $\endgroup$
    – Silas
    Feb 3, 2022 at 18:34
  • $\begingroup$ Write the interaction Hamiltonian between atom and photon. The conservation (selection rule) is the result of interaction Hamiltonian. $\endgroup$
    – ytlu
    Feb 3, 2022 at 20:22
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    $\begingroup$ The reasoning for the selection rules for $\Delta m$ is similar as that for $\Delta\ell$. The angular momentum of the photon also has projections. For example $q_\mathrm{ph}=0$ and $\pm 1$ are associated with linearly polarized light and left/right circular polarized light. If you have non polarized light, you have to sum over all contributions of the projection quantum number. To get better intuitive understanding you need to know some angular momentum algebra (the selection rules follow directly from the 3j Symbols). $\endgroup$
    – Paul
    Feb 4, 2022 at 19:21

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The selection rule for $m$ corresponds to the conservation of the $J_z$ component of the angular momentum, where $z$ is the chosen quantization axis.

  • The $\Delta m=0$ possibility occurs when the system is driven by light which is linearly polarized along the quantization axis, in which case each photon has total angular momentum $1$ but nevertheless has $J_z=0$.

  • The $\Delta m=\pm1$ case happens when the system is driven by light which is circularly polarized in the plane orthogonal to the quantization axis, in which case each photon has angular momentum component $J_z=\pm 1$ that must be absorbed by the system.

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