Why is the Sky Blue? Understanding Born's Rule & Rayleigh Scattering

[Agrippa]

I recently asked a physics professor to cite some observable natural phenomena whose explanation essentially requires Born's rule. He cited the sky's being blue. But I've had some trouble confirming this through online searches.

Typically one finds the claim that the blue colour of the sky is caused by the scattering of sunlight off the molecules of the atmosphere. This so-called 'Rayleigh scattering' is more effective at short wavelengths (the blue end of the visible spectrum). So the light scattered down to the Earth at a large angle with respect to the direction of the sun's light is predominantly in the blue end of the spectrum.

Where exactly does the Born rule fit into that standard type of explanation?

 

[Jilang]

The probability of observing a blue photon will involve Borns rule and and integration over those wavelengths we call blue.

 

[Agrippa]

The probability of observing a blue photon will involve Borns rule

Yes of course, I'm looking for the mechanism underlying that claim.
I understand that Born's rule states that the probability of observing a blue photon is proportional to the squared absolute value, of the coefficient that the quantum state attributes to observing a blue photon. But what's the relevant quantum state?

and integration over those wavelengths we call blue.

I understand that colour is a property of objects that disposes them to reflect light at certain wavelengths. So we call the sky "blue" because atmospheric molecules reflect light at "blue" wavelengths. Still, what is actually happening? Are the photons actually in a superposition of being reflected at "blue" wavelengths, "red" wavelengths (etc.) such that the squared absolute value of the blue coefficient is relatively very high? If so WHY is blue light amplitude so high?

 

[Nugatory]

I recently asked a physics professor to cite some observable natural phenomena whose explanation essentially requires Born's rule. He cited the sky's being blue.

Hmmm... My first thought was that this is SO not an example that I would have chosen... But it's surprisingly hard to find a good example. Single photons and stacked polarizing filters perhaps? You asked for "some" examples - did your professor have more than this one?

 

[morrobay]

Yes of course, I'm looking for the mechanism underlying that claim.
I understand that Born's rule states that the probability of observing a blue photon is proportional to the squared absolute value, of the coefficient that the quantum state attributes to observing a blue photon. But what's the relevant quantum state?I understand that colour is a property of objects that disposes them to reflect light at certain wavelengths. So we call the sky "blue" because atmospheric molecules reflect light at "blue" wavelengths. Still, what is actually happening? Are the photons actually in a superposition of being reflected at "blue" wavelengths, "red" wavelengths (etc.) such that the squared absolute value of the blue coefficient is relatively very high? If so WHY is blue light amplitude so high?

What actually is happening ? The mechanism is classical :
The natural resonant frequency of the electrons in the N₂ and O₂ most closely corresponds to the frequency of blue and UV light.
Therefore the system in the visible range is "driven" most effectively by blue light ( absorbed and re emitted - scattered)

 

[Agrippa]

Hmmm... My first thought was that this is SO not an example that I would have chosen... But it's surprisingly hard to find a good example.

Yes, that's what I've been discovering, and it is rather surprising!

Single photons and stacked polarizing filters perhaps?

Not exactly sure what you're suggesting here - a Born rule based explanation of glare reduction?

You asked for "some" examples - did your professor have more than this one?

Other than a brief mention of thermal properties of matter (since thermodynamics is apparently ultimately to be derived from QM with Born rule), no he didn't.
After some googling, it looked like the best examples might come from quantum biology. I'm thinking of using the example of plant growth, since the efficiency of photosynthesis in plants appears to be explained by the fact that the most efficient path of energy transfer, to the reaction center of a plant, is the high amplitude path. But I can't find any justification for the crucial claim that the efficient path is the high amplitude path - what's stopping the least efficient path from being the high amplitude path?
(It's analogous to the problem of the why the blue wave-length should get the high amplitude.)

 

[Agrippa]

What actually is happening ? The mechanism is classical :
The natural resonant frequency of the electrons in the N₂ and O₂ most closely corresponds to the frequency of blue and UV light.
Therefore the system in the visible range is "driven" most effectively by blue light ( absorbed and re emitted - scattered)

Are you sure about that? This article suggests the complete opposite, it says:

"Reflection and transmission of light waves occur because the frequencies of the light waves do not match the natural frequencies of vibration of the objects. When light waves of these frequencies strike an object, the electrons in the atoms of the object begin vibrating. But instead of vibrating in resonance at a large amplitude, the electrons vibrate for brief periods of time with small amplitudes of vibration; then the energy is reemitted as a light wave."

But perhaps your point about this being non-quantum may still follow. The idea would be that light in the frequency interval 610–670 THz (i.e. blue light) is the only light that does not correspond (whatever "correspond" means) to the natural resonant frequency of the electrons of atmospheric molecules such that blue light is the only light that these molecules spit back out; and that process is described entirely by classical physics.

But I don't see why this process is described entirely by classical physics. For aren't photons typically in superposition of different frequencies? https://www.researchgate.net/post/Can_a_photon_exhibit_multiple_frequencies . But then your claim that the mechanism is classical doesn't seem right, and one wonders why the non-resonating frequency of the photon is the frequency with the most Born rule probability.

 

[morrobay]

See section 1138 http://farside.ph.utexas.edu/teaching/em/lectures/node97.html
The basic mechanism in Rayleigh scattering is frequency dependent : α 1/λ⁴ where the incident time varying E field induces a time varying dipole moment in N₂ and O₂ that radiate (scatter) light)
In QED scattering is a two step process : absorption and emission and the amount of scattering is smaller away from resonance.
The Borne rule is related to most likely outcome /probability.
And again electrons in atoms and molecules are bound by strong restoring forces and have definite natural frequencies, that correspond to the violet and UV region. Blue light is closest to the natural resonant frequency of the bound electrons.
So the time varying electric field vector in the blue incident wave is more effective in causing forced oscillations
( dipole moments ) in atmospheric N₂ and O₂
I will let one of the physicists here elaborate on the exact relationship between absorption/re emission
and an induced time varying dipole moment.

 

[Agrippa]

See section 1138 http://farside.ph.utexas.edu/teaching/em/lectures/node97.html
The basic mechanism in Rayleigh scattering is frequency dependent : α 1/λ⁴ where the incident time varying E field induces a time varying dipole moment in N₂ and O₂ that radiate (scatter) light)
In QED scattering is a two step process : absorption and emission and the amount of scattering is smaller away from resonance.
The Borne rule is related to most likely outcome /probability.
And again electrons in atoms and molecules are bound by strong restoring forces and have definite natural frequencies, that correspond to the violet and UV region. Blue light is closest to the natural resonant frequency of the bound electrons.
So the time varying electric field vector in the blue incident wave is more effective in causing forced oscillations
( dipole moments ) in atmospheric N₂ and O₂
I will let one of the physicists here elaborate on the exact relationship between absorption/re emission
and an induced time varying dipole moment.

Thanks, this is very helpful. However, I think the link you provide goes against the claim that the mechanism is entirely classical, and at least points us in the direction of an answer to my original question. In particular, the link admits that "Indeed, we are very crudely modeling our hydrogen atom".

The key point is that there are semiclassical descriptions of the atom, no classical description of atom, and the semiclassical description is just an approximation of scattering theory, which is quantum.

You've explained that electrons in atoms and molecules are bound by strong restoring forces and have definite natural resonant frequencies, that correspond to the violet and UV region. Blue light is closest to the natural resonant frequency of the bound electrons. But I think the more accurate quantum description will entail that the electrons are instead in superposition of different frequencies such that the frequency corresponding to blue light has (extremely) high mod-square amplitude. If that's right, then that's how the Born rule ultimately fits into the explanation of the sky being blue.

 

[Jilang]

Perhaps he was thinking of the probability of transitioning between states which has a similar form to the born rule? The density of states due to the different vibrational modes increases with frequency.