Trouble understanding how charge density is the amplitude squared

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A little while ago I posted a thread asking the same question but didn't really get an answer so I am going to try to rewrite it. I'm having trouble understanding how the charge density of an electron (or any particle) is the square of its wave function. One problem I have is when there are two protons and one electron who's amplitude is evenly spread on both protons. It seems to me that if the charge density is the square of the amplitude the charge of the electron should be split between both protons and they should feel a repulsive force because each proton is only half shielded. But I was told here and in the Feynman Lectures in physics that they don't repel for this reason because one proton is shielded (or screened). I haven't read any material that throughly discussed the charge density being the square of the amplitude so that might be my problem, but can anyone help me understand this?

 

[MalleusScientiarum]

The square of the wave function is the probability of finding a particle between the points x and x + dx, right? Let's suppose I want to find the average charge between x and x + dx, then. I would expect it to be the charge of the particle times the probability of finding the particle in that region, on average, right? Well, in that case we can interpret the square of the wave function as being the way that a fake charge is "distributed" through space.

 

[Nicky]

[...] One problem I have is when there are two protons and one electron who's amplitude is evenly spread on both protons. It seems to me that if the charge density is the square of the amplitude the charge of the electron should be split between both protons and they should feel a repulsive force because each proton is only half shielded. But I was told here and in the Feynman Lectures in physics that they don't repel for this reason because one proton is shielded (or screened). [...]

If you are talking about an ionized hydrogen molecule (H2+), the electron's charge density is mostly between the two protons. You could think of the system as three point charges along a line:

(H+)===(e-)===(H+)

From each proton's perspective, the electron is closer than the other proton, so the electron's attractive force is stronger than the other proton's repulsive force.