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

TL;DR

This paper introduces a Madelung fluid model for a free particle in two dimensions, deriving a class of stationary, localized, and spinning wave functions that generalize the Schrödinger equation by including rotational quantum flows.

Contribution

It presents a novel Madelung fluid framework allowing rotational flows and characterizes the most likely wave functions with fixed average energy, including self-trapped and spinning solutions.

Findings

Existence of stationary, localized, spinning wave functions.

Balance between quantum potential and centrifugal force.

Special non-spinning stationary state as the lowest energy solution.

Abstract

We consider spatially two dimensional Madelung fluid whose irrotational motion reduces into the Schr\"odinger 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 Schr\"odinger equation for a particle in a cylindrical tube classical potential.