〈 Berger | Dillon 〉

Also, Coulomb assume point charges. This means well localized. But within an atom, that picture fails.

it's a perfectly fine approximation to ignore radius of the proton.. and just because a particle is a point particle doesn't mean that the point has a well defined, localized position..

True. But then, what does it means that r in U?

you could just as well ask what does the x,y,z mean in *any* potential V(r) mean.. They are just the coordinates or the electron--whatever they may be..

Perhaps is just the language. My fault that I didn't thought in these terms for a while. But such questions came to me few days ago. You just gave me the opportunity to ask you. :)

If one says U = e^2/r, one thinks on 1) a given position 2) a localized charge. which is usually not always the case the case in QM, not even in a bound state like an atom.

but all of that is covered from the fact that its wave function must obey the wave equation.. if you ask where the particle is you have to integrate it against the probability distribution defined by the schrodinger wave equation, where the coordinates are treated as random vars

Of course, but one could say that that is a posteriori.