Fall of Quantum Particle to the Center: Exact results

The fall of a particle to the center of a singular potential U(r) is one of a few fundamental problems of quantum mechanics. Nonetheless, its solution is not complete yet. The known results just indicate that if U(r) decays fast enough at r tends to zero, the spectrum of the Schrodinger equation is not bounded from below. However, the wave functions of the problem are singular at r = 0 and do not admit the limiting transition to the wave function of the ground state. Therefore, the unboundedness of the spectrum is only the 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. Specifically, an exact solution of the time-dependent Schrodinger equation describing the fall to the center is obtained and analyzed. The law describing the collapse to a single point of the region of the wave function localization is obtained explicitly, as well as the temporal dependences of the average kinetic and potential energy of the falling particle and its momentum. It is shown that the known necessary conditions for the particle to fall simultaneously are sufficient.