Is there a quantum mechanical explanation of light scattering by atmospheric gases? Classically, treating the atmospheric particles as excited dipole antennas, we know that the amount of scattering is inversely proportional to the fourth power of the wavelength. Is there a quantum mechanical explanation for this scattering profile perhaps in terms of electronic excitations?
Yes, there is a QM level scattering.
When a photon interacts with an atom, three things can happen:
elastic scattering, the photon keeps its energy and changes the angle
inelastic scattering, the photon gives part of its energy to the atom and changes the angle
absorption, the photon gives all its energy to the atom/electron
In your case, Rayleigh scattering is 1., elastic scattering. This is the only way the photons' energy and relative phase is kept and the images through the air (homogeneous) are transferred without bigger distortion.
By the way, this is the reason why the sky is blue, but not as simple as you would think. As the light is white (yes it is white and not yellow) coming from the sun, it is made up of combinations of all color photons. Now as white light scatters elastically on the atoms of the air, Rayleigh scattering is dependent on the wavelength strongly. The particles that photons scatter on (air atoms) need to be much smaller than the wavelength of the photon to have Rayleigh scattering. Now, this is why shorter wavelength photons (blue) will dominate the sky. Blue wavelength photons will be scattered more dominantly (elastically) then longer wavelength photons, so from all directions, the indirectly coming photons look blue.
Now if you look directly to the Sun, you will see direct photons coming from the Sun, dominated by ones that are not scattered elastically, and these will appear in the longer wavelength (Yellow).
Just to make it clear, all three happens on air molecules, elastic, inelastic and absorption too. But the elastic scattering is what dominates and that transfers images undistorted through the air, and the other two, inelastic scattering and absorption is just happening in a much smaller ratio. Inelastic scattering is Compton/Thomson scattering which happens too. Absorption too, that heats up the air (this is the one where electrons get excited).
You are talking about the electronic excitations, and contrary to popular belief on this site, the elastically scattered photons that make the sky look blue are not absorbed nor re-emitted.
From Wikipedia (https://en.wikipedia.org/wiki/Rayleigh_scattering):
This radiation is an integral part of the photon and no excitation or deexcitation occurs.
So the QM explanation is that the photons that cause the blue sky, which are elastically scattered, are not absorbed, they do not cause the electrons around the atom to be excited. The photon elastically scatters off the atom, which means that the kinetic energy of the particle is conserved in the center of mass-frame.
In Rayleigh scattering a photon penetrates into a medium composed of particles whose sizes are much smaller than the wavelength of the incident photon. In this scattering process, the energy (and therefore the wavelength) of the incident photon is conserved and only its direction is changed. In this case, the scattering intensity is proportional to the fourth power of the reciprocal wavelength of the incident photon.
Now your question could be ok, but how does then the photon change direction? This is a very complex thing to explain, and involves the wave nature of the photon in QM, and first you need to understand how the photons as partial waves travel through the slits in the double slit experiment and interfere with themselves just to change the direction of the photon and cause a random place on the screen for the photon to be absorbed, but that random place still creates an interference pattern when this is repeated many times. Analogously, the photons scatter in the air molecules (like the slits) as they travel through the air as partial waves just to interfere with themselves and change the angle.
You can see the same change of angle when photons travel from one medium to another (or in the same medium the medium changes density or other characteristics), the photons on the lattice of the new medium will scatter elastically and interfere with themselves as partial waves and change the angle. This is the QM explanation of the refraction, again contrary to popular belief on this site, this is what causes in QM level the photons to change the angle. They are interacting with the center of mass frame (the atom).