Free-fall in a uniform gravitational field in non-commutative quantum mechanics

TL;DR

This paper investigates the effects of noncommutative quantum mechanics on free-fall in a gravitational field, deriving bounds from experiments and extending the equivalence principle within this theoretical framework.

Contribution

It provides an exact solution for a quantum particle in NCQM under gravity, constrains noncommutativity parameters using experimental data, and extends the equivalence principle to NCQM.

Findings

Upper bound on noncommutativity parameter from GRANIT experiment

Time of flight matches classical results in NCQM for far-from-turning-point measurements

Extension of the equivalence principle to noncommutative quantum mechanics

Abstract

We study the free-fall of a quantum particle in the context of noncommutative quantum mechanics (NCQM). Assuming noncommutativity of the canonical type between the coordinates of a two-dimensional configuration space, we consider a neutral particle trapped in a gravitational well and exactly solve the energy eigenvalue problem. By resorting to experimental data from the GRANIT experiment, in which the first energy levels of freely falling quantum ultracold neutrons were determined, we impose an upper-bound on the noncommutativity parameter. We also investigate the time of flight of a quantum particle moving in a uniform gravitational field in NCQM. This is related to the weak equivalence principle. As we consider stationary, energy eigenstates, i.e., delocalized states, the time of flight must be measured by a quantum clock, suitably coupled to the particle. By considering the clock as a small perturbation, we solve the (stationary) scattering problem associated and show that the time of flight is equal to the classical result, when the measurement is made far from the turning point. This result is interpreted as an extension of the equivalence principle to the realm of NCQM.