paper

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

arXiv:2202.12615 · doi:10.1098/rspa.2023.0366

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 Schrödinger'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.

arXiv comment11 pages 2 figure. arXiv admin note: text overlap with arXiv:2108.01981

References in corpus (12)