Zero absolute vorticity plane Couette flow as an hydrodynamic representation of quantum energy states under perpendicular magnetic field

Abstract

Here we extend the Madelung transformation of the Schr\"{o}dinger equation into a fluid-like form to include the influence of an external electromagnetic field on a charged particle. The vorticity of the Madelung fluid is then in the opposite direction to the imposed magnetic field and equal in magnitude to the cyclotron angular frequency. When the particle motion is confined to a plane, perpendicular to an imposed magnetic field, the equivalent flow dynamics is that of zero absolute vorticity obtained in a quasi 2D rotating frame, where the cyclotron frequency plays a role equivalent to that of the Coriolis frequency in a rotating frame. We show how the Landau levels and the extended modes in the integer quantum Hall effect are all mapped into such zero absolute vorticity-like plane Couette flows, where the latter exhibit a geostrophic-like balance between the magnetic force and the gradients of the quantum (Bohm) potential and the electric force.