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The selection rules have $\Delta m =0$ for light polarised in the z-direciton trying to excite an atom. Mathematically it is clear that $\langle n, l, m_l| z |n, l, m_l \rangle = 0$ unless $\Delta m=0$. Therefore I understand the mathematical origin of this fact, I'm just looking for the most intuitive physical description of why this is true.

I have thought of the fact that there are no $x$,$y$ terms in the electric dipole radiation term in the Hamiltonian means that $m_l$ must be conserved through these transitions, but again this seems more 'mathematical' than intuitive.

I guess maybe it could just be geometrically obvious that any direction that is distorted solely by its z-coordinate cannot have its direction with respect to the z-axis affected?

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    $\begingroup$ The classical explanation is just that you need a torque about the z-axis in order to change the angular momentum about that axis, but a force parallel to the z-axis can't produce a torque about the z-axis. $\endgroup$
    – Paul G
    Feb 22, 2021 at 1:29
  • $\begingroup$ Yeah that makes sense to me, thanks! $\endgroup$
    – Alex Gower
    Feb 22, 2021 at 11:55

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As @Paul G said, clasically the explanation is that you require a torque about the z-axis in order to change the angular momentum about that axis. Therefore a force parallel to the z-axis cannot produce a torque about the z-axis.

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