I ran across a neat quantum mechanics question recently:
Consider a scattering gedankenexperiment in which spinless particles of a given energy are directed at a target. We wish to measure the wave function downstream from the scatter. We assume the beam is described by a pure state, and that the incident wave is a plane wave (over a sufficiently large spatial region). The beam is low density, so the particles do not interact with one another. To measure $|\psi(r)|^2$ over some volume of space, we just put a screen $S$ in a certain location, and gather enough statistics to get the probability density on this surface. We then move the surface and measure again.
Describe a modification by which the phase of the wave function $\psi(r)$ can be measured on the screen, apart from the overall phase, which is nonphysical and can never be measured.
I imagine it can be possible to infer the phase by seeing how the scattered wavefunction interferes with a "reference" wave, which can be done by, e.g. splitting off a piece of the incoming wave and routing it around the target to the desired point. But this feels a bit convoluted. Is there a nice and direct way to measure the phase of the scattered wave?