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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