Eur. Phys. J. D (2016) 70: 267
https://doi.org/10.1140/epjd/e2016-70463-3

Regular Article

Falling of a quantum particle in an inverse square attractive potential

Department for Theoretical Physics, Ivan Franko National University of Lviv, 12, Drahomanov St., 79005 Lviv, Ukraine

^a e-mail: waswasiuta@gmail.com

Received: 21 July 2016
Received in final form: 5 October 2016
Published online: 20 December 2016

Abstract

Evolution of a quantum particle in an inverse square potential is studied by analysis of the equation of motion for 〈r²〉. In such a way we identify the conditions of falling of a particle into the center. We demonstrate the existence of a purely quantum limit of falling, namely, a particle does not fall, when the coupling constant is smaller than a certain critical value. Also the time of falling of a particle into the center is estimated. Although there are no stationary energy levels for this potential, we show that there are quasi-stationary states which evolve with 〈r²〉 being constant in time. Our results are compared with measurements of neutral atoms falling in the electric field of a charged wire. Modifications of the experiment, which may help in observing quantum limit of falling, are proposed.