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Madelung Fluid Model for The Most Likely Wave Function of a Single Free Particle in Two Dimensional Space with a Given Average Energy

We consider spatially two dimensional Madelung fluid whose irrotational motion reduces into the Schrödinger equation for a single free particle. In this respect, we regard the former as a direct generalization of the latter, allowing a rotational quantum flow. We then ask for the most likely wave function possessing a given average energy by maximizing the Shannon information entropy over the quantum probability density. We show that there exists a class of solutions in which the wave function is self-trapped, rotationally symmetric, spatially localized with finite support, and spinning around its center, yet stationary. The stationarity comes from the balance between the attractive quantum force field of a trapping quantum potential generated by quantum probability density and the repulsive centrifugal force of a rotating velocity vector field. We further show that there is a limiting case where the wave function is non-spinning and yet still stationary. This special state turns out to be the lowest stationary state of the ordinary Schrödinger equation for a particle in a cylindrical tube classical potential.