from the fact that you have two charged particles, the electron and the proton
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Of course, but one could say that that is a posteriori.
i'm not sure what you mean.. the coordinates are treated as random variables of a distribution that need to be integrated over for meaningful results.. this isn't just physics.. just because a distribution f(x,y) has coordinates x,y doesn't mean x,y definitely happen
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I mean, one first establishes the Hamiltonian, which is a function, not a distribution and afterwards one writes down the Schrödinger eq. and voilá QM and so on. I'm thinking about the Hamitonian and the meaning of its terms, with QM. Perhaps my way of thinking is not well posed
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you can just think of the terms of the Hamiltonian in position space as H(x,y,z) such that H(x,y,z)|ψ|^2 d^3x is the average energy in a box of volume d^3x located at (x,y,z)
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