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Is there a charge density in quantum mechanics?

Is it meaningful in quantum mechanics to speak of charge distribution? Some people say if you sovle the Schroedinger Equation for the hydrogen atom, the eigenfunctions represent a type of charge density...that is, the square of the amplitude is a charge density. Yes, I know some people say it's not charge density, it's just the probability of finding an electron, but let's just call it charge density. That's not the point.

The point is: in more complex systems, does quantum mechanics recognize the existence of a similar quantity to the square of the amplitude in the hydrogen atom...a generalized charge density? or at least, a probability of finding charge. If I look at a glass of water, can I say that according to quantum mechanics, for any given point (x,y,z) there is a certain expectation dQ within any given volume element dV that there is a unit charge within that volume?

Or, as certain correspondents have claimed in a related question which was shut down by the moderators, is it meaningless in quantum mechanics to talk about such a quantity dQ/dV?

See http://physics.stackexchange.com/questions/241247/semi-classical-calculation-gives-wrong-answer-for-emission