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When a charged particle absorbs a photon, does the particle accelerate towards where the photon was coming from, or where the photon was going?

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If you mean actual photons, the particle gets pushed. Photons have momentum that points along their direction of travel.

You may be confused because you might have heard that the electromagnetic force comes from the exchange of photons. So if charges throw photons at each other, it looks like they should only be able to repel, and never attract. The difference is that here the photons are 'virtual', which means they can be 'off-shell' and do all kinds of weird things -- a virtual photon can move to the left while having momentum that points to the right, so it can pull!

What's actually going on is much more complicated and looks nothing like the exchange of particles at all, it's just that if you do some approximations you'll get terms in your expansion that look like particles are being exchanged. But they're these weird 'virtual' particles, not real ones.

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Photons are a kind of quanta so if they are interacting with something, that something is also a kind quanta.

A classical charge can have a location and a momentum it can look at the value of a classical field at the place it is and adjust it's momentum at a particular rate based on the direction that electromagnetic field points compared to its own motion.

But the quanta can't do that. The charge doesn't have a location and a momentum so its wouldn't know where to evaluate the field or what its momentum is to change the momentum. And the photon doesn't have a value for the electromagnetic field at that location away. So it's simply not an option, that isn't going to happen.

What you have is a photon field and an electron-positron field (they share a field in common because they are antiparticles). On their own they might evolve a certain wave, having modes and such. But they can interact with each other as well. And they do.

This interaction is sometimes similar to the classical case you ate familiar with, sometimes there are many modes with a large occupancy of the photon field that have a common phase that collectively act very similar to a classical electromagnetic wave. And the electron field might have a mode that in some respects looks similar to a charge moving along.

But they just have more possibilities in general. Which means you could have an interaction where you start with and electron and positron bound together similar to a hydrogen atom and there is some modes of the photon field corresponding to the excitation energy. Then it could evolve to the electrons positron being a higher energy bound state and the photon field has a made with one fewer quanta in its occupancy.

But the electron and positron were moving around at a location with a momentum, neither of them picked up some average momentum (and they have opposite charges anyway).

The electron and positron could have a mode that has a center of mass and a relative separation that has it's location of the center of mass spread out and has its relative separation spread out but strongly concentrated on them being not too far away from each other. After the interaction with the photon field it might instead be strongly in a mode where the location of the center of mass is still spread out but the relative separation is spread out more.

Just like a hydrogen atom can get a bigger average radius if it (the atom) absorbs a photon.

The charge didn't absorb the photon. The system of two charges which was already spread out and in a particular state where it could have different locations and different momentum changes to become a different state which again can have different momentum and different locations its the distribution that changes.

And since the distributions for position and momentum are for distributions for different interactions they are not a joint distribution so it is not like there are location, momentum pairs that then adjust as if they interacted with an electromagnetic field.

So, it's more complicated.

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