Classical Characterization of quantum waves: Comparison between the caustic and the zeros of the Madelung-Bohm potential

TL;DR

This paper compares classical wave descriptions via caustics and Madelung-Bohm potential zeros, revealing that fold caustic beams are the most classical, with implications for understanding wave behavior and superoscillations.

Contribution

It provides a geometric comparison between caustic-based and Madelung-Bohm potential-based classical descriptions of quantum waves, highlighting their relation in specific beam types.

Findings

Fold caustic beams have zeros of Madelung-Bohm potential coinciding with the caustic.

For Airy and Pearcey beams, caustics are stable and the descriptions align.

Zeros of Madelung-Bohm potential relate to superoscillation phenomena.

Abstract

From a geometric perspective, the caustic is the most classical description of a wavefunction since its evolution is governed by the Hamilton-Jacobi equation. On the other hand, according to the Madelung-de Broglie-Bohm equations, the most classical description of a solution to the Schr\"odinger equation is given by the zeros of the Madelung-Bohm potential. In this work, we compare these descriptions and, by analyzing how the rays are organized over the caustic, we find that the wavefunctions with fold caustic are the most classical beams because the zeros of the Madelung-Bohm potential coincide with the caustic. For another type of beams, the Madelung-Bohm potential is in general distinct to zero over the caustic. We have verified these results for the one-dimensional Airy and Pearcey beams, which accordingly to the catastrophe theory, their caustics are stable. Finally, we remark that for certain cases, the zeros of the Madelung-Bohm potential are linked with the superoscillation phenomenon.