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

Abstract: We propose to revisit a fundamental quantum problem, namely the evolution of an electron's wave function under a time-dependent magnetic field, with the dual perspective of the Madelung fluid. First we present an analysis of the problem in a quantum mechanistic fashion, based on a perturbation theory of the Landau levels, and next address the same problem with the Madelung equations. We show that the latter formulation does not only provide an intuitive derivation of the solution, but it also allows us to understand the diabatic character of quantum evolution in terms of mechanical energy transfers. The sloshing oscillations of the wave function can be then interpreted as the consequence of deviations from the balance between the magnetic force and the gradient of the Bohm potential in the Landau levels. This study shows that the Madelung fluid approach reveals analogies between a priori unrelated concepts from quantum mechanics and geophysical fluid dynamics.