Current conservation and amplitude regularisation of the Landau problem: Bohm--Madelung description

TL;DR

This paper explores the Bohm--Madelung formulation of the Landau problem, analyzing regularisation schemes that preserve spectral structure and provide a framework for stationary Bohmian dynamics in magnetic fields.

Contribution

It introduces and compares global Fisher-information and local canonical regularisation schemes for the Landau problem within the Bohm--Madelung framework, highlighting their structural effects.

Findings

Radial and axial sectors are globally regularisable, maintaining analytic structure.

Azimuthal sector exhibits complex amplitude structure due to gauge coupling.

Local regularity is achieved through amplitude--momentum relations at the canonical branch level.

Abstract

This work investigates the dynamics of a charged particle in a uniform magnetic field within the Bohm--Madelung formulation of quantum mechanics. In this representation, the stationary Schrodinger equation separates into coupled amplitude and phase equations, where the amplitude sector admits a Sturm--Liouville structure supporting Ermakov--Lewis invariants. The analysis considers two complementary regularisation schemes: a global Fisher--information--based regularisation and a local canonical (shell) Bohm regularisation derived from stationary flux closure. These are applied within distinct classes of stationary flow, characterised by vanishing and nonvanishing current components. It is shown that the radial and axial sectors remain globally regularisable, preserving analytic structure across the domain. In contrast, the azimuthal sector develops a nonseparable, generally complex-valued amplitude structure due to gauge-induced coupling. Nevertheless, a consistent local regularity is recovered at the level of canonical branches, where amplitude--momentum relations organise the solution in a well-defined manner. Regularisation thus acts as a structural reorganisation mechanism in amplitude space, preserving the Landau spectral scale while reorganising the flux-sector structure through branch-wise amplitude--momentum relations, thereby establishing a natural framework for the description of stationary Bohmian dynamics in the Landau problem.