quantum mechanics - How can the phase of a wavefunction be measured?

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?

Answer

I'll describe here a conceptual method for phase measurement without going into the details of the experimental complications, the conditions of validity, or the accuracy.

Given a system of particles of mass $m$ described by the wave function in a stationary state. $$\psi(\mathbf{r}) = A(\mathbf{r}) e^{i\phi(\mathbf{r}) }$$ The probability current is given by: $$\mathbf{J} = \frac{\hbar}{2 mi } (\psi^{\dagger} \mathbf{\nabla} \psi - \psi \mathbf{\nabla} \psi^{\dagger} ) = \frac{\hbar}{ m } A(\mathbf{r}) \mathbf{\nabla} \phi(\mathbf{r})$$ Suppose that you have very small directional sensors, which can measure the intensity perpendicular to their surface, and a very precise timer. Then in principle you can put sensors along the coordinate axes in the vicinity of some point in space, and measure the currents which will be proportional to $\frac{\partial\phi}{\partial x}$, $\frac{\partial\phi}{\partial y}$, $\frac{\partial\phi}{\partial z}$. These measurements will allow the construction of the phase up to an additive constant.

The measurement results will be proportional to the square root of the intensity: $\sqrt{|A|^2}$; and in order to remove this factor, we will need an additional volumetric sensor to measure the intensity in a small volume around the same point in space.

This principle is the basis of the widely known noninterferrometric phase measurement (or phase retrieval\reconstruction) techniques.