What is the electric field around a quantum particle?

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Beginning another attempt in my amateurish efforts to understand quantum mechanics, I suddenly realized there was a basic issue I find completely unclear.

When we solve the SE for an interesting system we first need to input a potential, which is usually some function of the spatial coordinates. For example courses often start by looking at a particle in a box, and then of course the hydrogen atom. But when we solve the hydrogen atom as a single particle state in a fixed Coloumb potential this must just be an approximation. Since in fact the thing it's orbiting must also be represented by a wavefunction. So in fact we need a multi-particle state. Now I know we must be able to deal with multi-particle states since I have vague memories knocking around of doing the helium atom using perturbation theory, but I'm having some conceptual trouble getting my head around the nature of the electric field around things that don't have a definite position. Is the E-field at some point in space also a quantum observable with some probability of being this or this, depending on the different weighting of all the combinations of possible positions of all charged particles?
 

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