How does gravity affect the wavefunction of a particle? I'm wondering how gravity affects the wave function of a particle. For example, if we shot a particle horizontal to the earth at a vertical detector screen, would the distribution on the screen be ever so slightly elongated in the direction towards the earth, given that gravity's influence would be greater (even if by very small amounts) on the parts of the wave function closer to earth?
If gravity does indeed affect a particle's wave function, how does it affect it?
I see two possibilities:


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*The system's mass is localized to the point-like entity that we measure.

*The system's mass is it's whole wave function, with it's probability density representing where most of the mass is located.
Both scenarios bring up some interesting questions:


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*If the entire mass of the system is the particle, does that mean measuring the system changes the location of it's mass?

*Where would the mass be if we didn't measure the particle?

*If the mass of the system is instead the wave function's probability density that would essentially mean the particle isn't the system's whole mass. For example two particles of the same mass could theoretically have the same gravitational attraction to object, despite being different distances from the object. The same could be said for different forces of gravity on identical mass particles in the same location.
 A: Yes. Scientists have performed experiments using neutrons in gravity to show that the paths neutrons take are both affected by gravity and move on paths of quantized energy:

Valery Nesvizhevsky of the Institute Laue-Langevin and colleagues found that cold neutrons moving in a gravitational field do not move smoothly but jump from one height to another, as predicted by quantum theory.

For your particular question, you're asking about the wave function being spread out by gravity, and that's what would happen as a result of tidal forces, but the effect would be incredibly small unless you could get the neutrons' wave functions to spread out over a very large scale. This is because tidal forces depend on whether the size of the object is large enough to sample areas with different strength gravity. 
Edit for the update: the update has changed the sense of the question from how a wave function is affected by gravity to how the shape of a wave function affects the field produced by that particle. Experimentally this is going to be almost impossible to answer. If you think of electromagnetism as an analogue for gravity, though, you can find the answers you seek in studies of the "shielding effect" whereby inner shell electrons partially shield the outer shell ones from the nucleus. Note that electrons don't self-shield, at least to leading order in quantum field theory, so any picture you figure out will have to take that into account.
A: Gravity affects an atom by slowing the emission process (e.g. in an atomic clock), where each tick of the clock is shortened (i.e. time dilates). The change in length can be explained by applying the equivalence principle to the coordinates of the electron between absorption and decay events as an acceleration.  The wave function is continuous in time and does not recognized the ticks as electron events so it cannot be used to explain the dilation of time in terms of coordinate values no matter how intense the field is.
