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?