An essay on semiclassical analysis for microlocal singularities, turbulence intensity and integration of singularities by Schrödinger equation in probabilistic behavior

Abstract

The Schrödinger equation governs the probabilistic behavior of quantum particles through the wave function. Microlocal singularities denote regions with significantly high probability density or abrupt changes therein. By visualizing the probability distribution in time and space, we discern regions with higher probability density, indicative of potential microlocal singularities. These regions probably correspond to areas with a greater probability of particle presence. Such analysis aligns with Theorem 1, predicting characteristics of microlocal singularities of wave functions. Furthermore, Theorem 2 postulates that semiclassical path integrals along these singularities contribute significantly to solving the Schrödinger equation. Interpreting the temporal evolution of the probability density in the probability distribution visualization reveals the propagation of the particle over time. Regions of high density mean likely presence of particles at specific times, aligning with the predictions of Theorem 2. Consequently, the analysis of the contribution of high-density regions to the temporal evolution of the wave function resembles semi-classical path integral calculations. Thus, our findings demonstrate that visualization of probability distributions obtained from the numerical resolution of the Schrödinger equation allows a comprehensive interpretation of the behavior of quantum particles, consistent with the theorems.