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

TL;DR

This paper derives an exact equation of motion for a quantum particle in an inverse square potential, identifying conditions for quantum fall and the existence of a quantum limit, with implications for experiments involving neutral atoms.

Contribution

It provides an exact solution for the particle's fall time and demonstrates the quantum limit of falling based on the coupling constant, advancing understanding of quantum dynamics in inverse square potentials.

Findings

Existence of a quantum limit of falling below a critical coupling constant.

Derived exact equation of motion for $ extless r^2 extgreater$.

Proposed experimental modifications to observe quantum fall limit.

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.