Classical characterization of quantum waves: comparison between the caustic and the zeros of the Madelung–Bohm potential

Abstract

From a geometric perspective, the caustic is the most classical description of a wave function 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 Schrödinger 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 wave functions with fold caustic are the most classical beams because the zeros of the Madelung–Bohm potential coincide with the caustic. For another type of beam, 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, according to the catastrophe theory, have stable caustics. Similarly, we introduce the optical Madelung–Bohm potential, and we show that if the optical beam has a caustic of the fold type, then its zeros coincide with the caustic. We have verified this fact for the Bessel beams of nonzero order. Finally, we remark that for certain cases, the zeros of the Madelung–Bohm potential are linked with the superoscillation phenomenon.

Quantum potential in time-dependent supersymmetric quantum mechanics

“…However Planck's constant is very small, but not zero, and this term can be regarded as an additional self-generated potential that the classical particle experiences to give it its quantum nature. Furthermore, a number of authors have argued that a quantum particle behaves at its most classical in places where the quantum potential is zero [8][9][10]. From the point of view of the current work, the key thing to note from this formalism can be seen in the second of equations ( 2).…”

“…al. [9] have shown that wavefunctions with fold caustics are the most classical because the zeros of the quantum potential coincide with the caustic and the evolution of the caustic is governed by the Hamilton-Jacobi equation. Berry [10] has shown that, for quantum wavepackets, the Bohm potential vanishes on the boundaries of regions where the oscillations become superoscillatory.…”

Algorithm for Solving a System of Coupled Nonlinear Schrödinger Equations by the Split-Step Method to Describe the Evolution of a High-Power Femtosecond Optical Pulse in an Optical Polarization Maintaining Fiber

“…When numerically solving the Madelung equations, difficulties arise with errors in the zeros region of wave function. Errors in the calculations make the relative phase undefined [56,57]. It requires the control under the calculation of changes in the phase of a complex envelope.…”

Superoscillations and the quantum potential *

“…The first concerns the Madelung-Bohm representation of the Schrödinger wavefunction [1,2], where the quantum and classical momenta are governed by equations differing only by the quantum potential Q(r). Recently, the zeros of Q(r) have been explored [3], to investigate the significance of these hypersurfaces (points in 1D, curves in 2D, etc) and their possible interpretation as the most classical places in the wave.…”