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

Exact solutions describing a fall of a particle to the center of a singular potential (collapse) in classical and quantum cases are obtained, inspected, and compared. The quantum and classical solutions exhibit striking similarities. In particular, at zero energy of the particle, both are self-similar, and their characteristic spatial scales obey the same temporal dependence. In addition to collapsing, exact expanding solutions, corresponding to the escape of the particle from the singularity, exist for both problems. The collapse and escape are independent and not linked to each other through the singularity. In the quantum collapse, the wave functions of the exact solutions have time-depended norms. This case requires reconsidering and generalizing the continuity equations and rules for mean value calculations. It has also been done. The analysis sheds new light on the fundamentals of quantum mechanics.