Landau levels in a time-dependent magnetic field: the Madelung fluid perspective

TL;DR

This paper explores the quantum dynamics of a charged particle in a time-dependent magnetic field using the Madelung fluid approach, offering intuitive insights into non-adiabatic effects and energy transfer mechanisms.

Contribution

It demonstrates how the Madelung fluid formulation provides a clear physical interpretation of non-adiabatic quantum evolution and reveals analogies with geophysical fluid dynamics.

Findings

Hydrodynamic formulation yields an intuitive derivation of exact solutions.

Provides a physical interpretation of non-adiabatic quantum evolution.

Shows analogies between quantum dynamics and geophysical fluid phenomena.

Abstract

We revisit the quantum dynamics of a charged particle in a time-dependent magnetic field, a fundamental problem exhibiting rich non-adiabatic behaviour, from the complementary perspective of the Madelung fluid formulation. We first analyse the system within standard quantum mechanics using perturbation theory around the Landau levels, and then address the same problem through the Madelung perspective. We show that the hydrodynamic formulation not only yields an intuitive derivation of the exact solution, it also provides a clear physical interpretation of non-adiabatic quantum evolution in terms of mechanical energy transfers. In this picture, the sloshing oscillations of the wave function arise from deviations from the force balance between the magnetic Lorentz force and the gradient of the Bohm potential within the Landau levels. More broadly, our study illustrates how the Madelung approach reveals unexpected analogies between quantum dynamics and phenomena familiar from geophysical fluid dynamics.