Quantum Hydrodynamics: Kirchhoff Equations

TL;DR

This paper derives Kirchhoff equations from the Schrödinger equation, showing they describe quantum particle and wave behavior in two dimensions, and applies them to vortices and optical systems.

Contribution

It introduces a novel derivation of Kirchhoff equations from quantum mechanics and demonstrates their application to vortices and optical vortices.

Findings

Kirchhoff equations derived from Schrödinger equation.

Kirchhoff equations describe quantum particle and wave duality.

Application to optical vortices using paraxial approximation.

Abstract

In this paper, we show that the Kirchhoff equations are derived from the Schr\"odinger equation by assuming the wave function to be a polynomial like solution. These Kirchhoff equations describe the evolution of $n$ point vortices in hydrodynamics. In two dimensions, Kirchhoff equations are used to demonstrate the solution to single particle Laughlin wave function as complex Hermite polynomials. We also show that the equation for optical vortices, a two dimentional system, is derived from Kirchhoff equation by using paraxial wave approximation. These Kirchhoff equations satisfy a Poisson bracket relationship in phase space which is identical to the Heisenberg uncertainty relationship. Therefore, we conclude that being classical equations, the Kirchhoff equations, describe both a particle and a wave nature of single particle quantum mechanics in two dimensions.