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.

中文翻译：

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量子波的经典表征：马德隆-玻姆势的苛性碱和零之间的比较

从几何角度看，苛性碱是波动函数的最经典描述，因为其演化受汉密尔顿-雅各比方程式控制。另一方面，根据Madelung-de Broglie-Bohm方程，对Schrödinger方程解的最经典描述是由Madelung-Bohm电位的零给出的。在这项工作中，我们比较了这些描述，并且通过分析射线在苛性碱上的组织方式，我们发现具有苛性碱折叠的波函数是最经典的光束，因为马德隆-玻姆势的零点与苛性碱重合。对于另一种类型的光束，在苛性碱溶液中，马德隆-玻姆电位通常为零。我们已经验证了针对一维Airy和Pearcey光束的这些结果，根据突变理论，具有稳定的腐蚀性。同样，我们引入了光学的Madelung-Bohm电位，并且我们证明了，如果光束具有折叠类型的苛性碱，则其零点与苛性碱重合。我们已经验证了非零阶贝塞尔光束的这一事实。最后，我们指出，在某些情况下，马德隆-玻姆电位的零与超振荡现象有关。