Here is how I saw the Hamiltonian being written in one Quantum Mechanics book:
Well, That's all very nice and complicated, but looking at the last term, and seeing the use of $r$ in there,
creates a small doubt in me - what is this $r$ there?
In electrostatics, this was simply the distance between the two charged particles, and that would mean that this was the distance between the proton and the electron, but isn't the position of the electron non-deterministic? Shouldn't that thing be a probability distribution of the distance instead? And why isn't it squared?