So, I know experimentally that matter and light interact. And I can calculate cross sections in Quantum Electrodynamics. Yet a fundamental intuition about how light and matter interact is completely missing in my understanding.
If a photon were (for a brief instant) next to a proton, would they attract? Repel? How about an electron? Is there some kind of potential $V(x_2-x_1)$ which can approximate the interaction between light in a particle-like state and matter? If we can do it for protons and electrons, which are both particles and waves, why can't we also do it for photons?
Maybe this is going to be categorized as too broad. If that's so, let's just narrow it down to: Does a photon (a gaussian wavepacket of light) attract a proton, or repel it? If the answer depends on the distance, is there must be an optimal distance between them like in the Van der Waals potential?:
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