Time of falling of a quantum particle into an inverse square potential

Abstract

Evolution of a particle in an inverse square potential is studied. We derive an equation of motion for $\left<r^2\right>$ and solve it exactly. It gives us a possibility to identify the conditions under which a falling of a quantum particle into an attractive centre is possible. We get the time of falling of a particle from an initial state into the centre. An example of a quasi-stationary state which evolves with $\left<r^2\right>$ being constant in time is given. We demonstrate the existence of quantum limit of falling, namely, a particle does not fall into the attractive centre, when coupling constant is smaller then some critical value. Our results are compared with experimental measurements of neutral atoms falling in the electric field of a charged wire. Moreover, we propose modifications of the experiment, which allow to observe quantum limit of falling.