How does scattering work?

Why is the sky blue?

I was always taught in high school that light with wavelength $\lambda$ acts like a little particle that wiggles up and down through space (in proportion to its magnitude). I was under the impression for quite some time that higher energies yield higher frequencies which yield more "jiggling" and thus more scattering in the atmosphere.

However, after learning quantum physics and relativity, it seems like this is very wrong. From relativity it seems like light can't be "vibrating" or "jiggling" along its path. Light must travel in a perfectly straight line: if it didn't, then either violet photons would travel faster than red photons, or violet light would appear to propagate slower than red light (because the violet light must travel a wavier path).

Further, quantum physics treats the quantum amplitude of a photon (as far as I can tell) as a sort of internal property. It certainly doesn't seem to be physically realized as displacement in space.

So my question is, how does light scatter? Intuitively, it seems like light with higher frequency jiggles more and thus scatters broader and easier, but this intuition seems to be incorrect. If amplitude really is a property internal to the photon, how does it affect how the photon scatters?

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–  Qmechanic Dec 30 '12 at 19:55

EDIT: This question discusses why the sky is blue with regards to classical rayleigh scattering.

Light does indeed travel in straight lines. However, this just indicates the direction of the movement of the wave as a whole. A light wave is an electromagnetic oscillation, and therefore higher frequencies will have quicker oscillations. The wave itself will be moving forward at the same speed, but there will be "more waviness" per unit length for a higher frequency wave than for a lower frequency wave.

If you switch from the so called "wave picture" to the so called "particle picture" (the shabbily named wave-particle duality) then the frequency $\nu$ of a photon is related to it's energy as given by $$E = h \nu$$

These photons have an intrinsic position-momentum uncertainty wherein their position at any given moment cannot be described to an arbitrarily small accuracy. It is the probability of finding the photon at a given place that you have referred to as the quantum amplitude of a photon.

The square of the quantum amplitude describes the probability of the photon (neglecting time dependence for the moment) of being in a particular place in space, or possessing a particular momentum etc., depending on what basis you are describing the system in. So for example, if $\psi(x)$ is a function describing the quantum amplitude of a particle at every point in space, the probability that the particle lies somewhere between $0$ and $x$ is given by $$P(x) = \int_{0}^{x} |\psi(x)|^2 dx$$

One cannot predict it's position more definitively than this probability.

Now that I've clarified what I mean by a quantum amplitude - There does exist scattering theory for quantum mechanics. This scattering theory deals with the problem of how these quantum amplitudes change when confronted by an external "potential". A potential can also be a potential as created by another particle, so it covers that case as well. A frequently used approximation to help ease of calculation is the Born approximation. It just so happens that even without the Born approximation, for a coloumb potential (which is any regular atom), the scattering problem is exactly solvable. And that solution yields the same results as Rutherford scattering. So quantum mechanically as well, everything is quite well explained.

EDIT 2: I'd earlier said that Rayleigh scattering was exactly solvable, but it was actually Rutherford scattering. Rayleigh scattering is a limit of Mie scattering which is valid for EM waves scattering of particles. However, I suspect that if you consider photons as particles that interact with the coulomb potential, rutherford scattering will be a valid quantum mechanical description of the phenomenon. I haven't been able to find any links with the relevant details worked out, but once I do I'll link them here.

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I find it disturbing that people are upvoting this answer. Surely the scattering of sunlight by molecules in the air has nothing to do with coulomb potentials. And there is no attempt to address the question of why the sky is blue. –  Marty Green Dec 31 '12 at 3:15
Actually, in the question the OP implies that he is already aware of classical rayleigh scattering, so I didn't include it in my answer. He very specifically asks for a quantum explanation. Atoms/molecules do have a coulomb potential to within the Born approximation, and there is no analog of "hard sphere scattering" in QM. All scattering happens on potentials. –  Kitchi Dec 31 '12 at 8:38
@MartyGreen, Kitchi is correct. The original motivation for my question stemmed from considering Rayleigh scattering, but I was wondering how a "particle" moving in a "straight line" can scatter as if it were waving all over the place. –  So8res Dec 31 '12 at 15:48

A classical explanation is that an electromagnetic wave incident on a molecule causes the electrons to oscillate, and they in turn emit radiation in different directions.

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