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

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.