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

Abstract

A fall of a particle to the center of a singular potential is one of a few fundamental problems of quantum mechanics. Nonetheless, its solution is not complete yet. The known results just indicate that if the singularity of the potential is strong enough, the spectrum of the Schrodinger equation is not bounded from below. However, the wave functions of the problem do not admit the limiting transition to the ground state. Therefore, the unboundedness of the spectrum is only a necessary condition. To prove that a quantum particle indeed can fall to the center, a wave function describing the fall should be obtained explicitly. This is done in the present paper. Firstly, classical collapse is inspected. Then, the quantum problem is analyzed. A family of exact self-similar solutions of the time-dependent Schrodinger equation corresponding to the fall is obtained and discussed. A law for the collapse of the region of the wave function localization to a single point is obtained explicitly. It is shown that the known necessary conditions for the particle to fall simultaneously are sufficient. Comparison of the classical and quantum collapse exhibits striking similarity of the two cases.