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:
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:
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.