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Electrons are incredibly small, and the "size" of photons (or the distance away from one required to not interact with it) is also incredibly small. Yet photons still interact with electrons all the time. Are they attracted to each other in some way, does it have to do with the wave of the electron being larger than the supposed size of the electron, are they actually very unlikely to interact but there is simply an incredibly large amount of them, or a combination of these? Or maybe something else?

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    $\begingroup$ Consider that the size of an optical photon (~$0.5\times 10^{-6}$ m) is much larger than the distance between atomic nuclei (so also larger than the distance between nearest neighbor electrons) in most solids. $\endgroup$ – The Photon May 3 '18 at 5:03
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    $\begingroup$ And that photons with size similar to the atomic spacing (x-rays) are most known because they do penetrate solids without interacting (much) with them. $\endgroup$ – The Photon May 3 '18 at 5:04
  • $\begingroup$ The "size" of a photon is given by the heisenberg uncertainty principle and it is a locus of most probable interaction :Δ(x)Δ(p)>h/2π . where p momentum x space coordinate. $\endgroup$ – anna v May 3 '18 at 5:52
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The idea that charged particles (like electrons) interact with electromagnetic radiation (consisting of photons) is a fundamental fact of nature. When an electrically charged particle is accellerated, it emits photons, and when a photon encounters an electrically charged particle, it can force it to accellerate. This is true even for single photons and single particles; the existence of the interaction does not rely fundamentally on the presence of large numbers of photons or charges.

Experimentally, the strength of this interaction is furnished by the so-called fine structure constant, which is a measure of how likely a given photon is to interact with a charge.

The detailed physical model of this interaction is called quantum electrodynamics, or QED. A good source book on QED for nonspecialists is Richard Feynman's "QED: The Strange Theory of Light And Matter".

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Another view:

Electrons and photons are quantum mechanical entities and have to obey the Heisenberg uncertainty relation

The position and momentum of a particle cannot be simultaneously measured with arbitrarily high precision. There is a minimum for the product of the uncertainties of these two measurements. There is likewise a minimum for the product of the uncertainties of the energy and time.

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This means that even though elementary particles are point particles, there is a locus where they have a probability to manifest, dependent on their momentum. This probability is given by the solution of the wave equation the particle obeys, and is a wavefunction. The deBroglie wavelength generally summarizes the quantum mechanical nature of particles.

This is particularly appropriate for comparison with photon wavelengths since for the photon, pc=E and a 1 eV photon is seen immediately to have a wavelength of 1240 nm.

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whereas for a one eV electron,

the associated DeBroglie wavelength is 1.23 nm, about a thousand times smaller than a 1 eV photon

Once the electron and the photon are in proximity where the wave nature dominates, the interaction will have a measurable probability, dependent on the boundary values of the problem.

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