Fall of a Particle to the Center of a Singular Potential: Classical vs. Quantum Exact Solutions

• URL: http://arxiv.org/abs/2202.12615v4
• Date: Thu, 14 Dec 2023 11:48:21 GMT
• Title: Fall of a Particle to the Center of a Singular Potential: Classical vs. Quantum Exact Solutions
• Authors: Michael I. Tribelsky
• Abstract summary: We inspect the quantum problem with the help of the conventional Schr"odinger's equation. Surprisingly, the quantum and classical solutions exhibit striking similarities.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Exact solutions describing a fall of a particle to the center of a non-regularized singular potential in classical and quantum cases are obtained and compared. We inspect the quantum problem with the help of the conventional Schr\"{o}dinger's equation. During the fall, the wave function spatial localization area contracts into a single zero-dimensional point. For the fall-admitting potentials, the Hamiltonian is non-Hermitian. Because of that, the wave function norm occurs time-dependent. It demands an extension to this case of the continuity equation and rules for mean value calculations. Surprisingly, the quantum and classical solutions exhibit striking similarities. In particular, both are self-similar at the particle energy equals zero. The characteristic spatial scales of the quantum and classical self-similar solutions obey the same temporal dependence. We present arguments indicating that these self-similar solutions are attractors to a broader class of solutions, describing the fall at finite energy of the particle.

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