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Suppose we have two electrons that apply a Coulomb force on each other. At large distances, we can consider the two electrons as point charges and the direction of the Coulomb force would be on the straight line connecting them. Now, if they are close enough that their distance is comparable to the radius of their "position clouds", can we still accurately determine the direction of the Coulomb force, or it would be kind of random with a probability distribution?

This can also be asked about the momentum direction of two atoms after a low-velocity collision, since the repulsion is in fact a Coulomb force applied by the nuclei and the electron clouds to each other.

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  • $\begingroup$ "position clouds"? $\endgroup$ – Avantgarde Apr 20 at 22:11
  • $\begingroup$ What I mean is that the position of the electron is not exactly determined. In an atom, it's a cloud of probability around the nucleus. For a confined electron, the positions the electron can be is determined by the squared amplitude of its wavefunction. If you suppose that the electron is a point charge, it's position would have different possibilities. Now my question was, if each of these possibilities correspond to a possibility for the direction of the Coulomb force. Simple. $\endgroup$ – Ali Lavasani Apr 20 at 22:15
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Yupp, it sure is!

This is (at least part of) the point of quantum field theories: they are basically what you get when you try to apply the same principles of quantum mechanics - which basically amount to emplacing a limit on information content - to spatially-pervasive fields like the electromagnetic field, as opposed to just particles, and thus allows one to say that the electric (and magnetic!) field near a charged particle, like an electron, is indeterminate just like its position.

The interesting part about this - which is actually a rather nice "boon", is that it turns out when we do this, the quantum-treated "fields" actually turn out to also be able to manifest as "particles", in that fields and particles end up as two sides of the same coin: we can thus even treat the electron as a disturbance in a field, and disturbances in the electromagnetic field, like light, become particles! (Thus, photons!) Mathematically, the two are related by basically "tilting your head" in a mathematical sense(*), what a mathematician calls a "change of basis". I would also go to suggest that this, perhaps, is a rather more faithful-to-theory way to present a "duality" involving particles than the so-called "wave-particle duality": it is a field-particle duality.

(*) At least for the non-interactive case, i.e. considering the fields separately - it's called the "Fock space" construction. We don't have a good maths to describe the interactive case, and instead must resort to hacks: while quantum field theory of a pure field is thus rather elegant, quantum field theory that is useful is actually a giant freakin' hack!

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