Describing quantum intereference with only currents and densities

Question

I know about and believe to understand the general wave equation based Kirchhoff diffraction formula, which in the Fraunhofer limit leads to a farfield complex wave function by Fourier transforming the complex wave function of the incident beam in the aperture.

I believe it should be possible to describe this phenomenon through only the density (wavefunction squared) and probability current (quantum mechanically). These two should be able to provide the phase and amplitude information necessary to interfere something.

What I'm looking for is a "hydrodynamic" theory of diffraction and/or interference, which I think should be very well possible.

Has this been done before? Am I barking up a wrong tree?

Answer 1

Yes indeed. This is called the '"Hydrodynamical formulation of quantum mechanics". Please, see , for example, the introductory section (pages 8-17) of the following dissertation by Toshiki Shimbori. In this formulation, the probability density and current density are interpreted as a fluid's density and hydrodynamical current density respectively. Quantum mechanical quantities are then interpreted as (or proportional to) hydrodynamical quantities, for example, the hydrodynamical velocity potential as the quantum phase and the vorticity as the angular momentum.