Minimal Length Uncertainty Relation and gravitational quantum well

TL;DR

This paper investigates how a minimal length uncertainty, introduced via a deformed Heisenberg algebra, affects the energy spectrum of a particle in a gravitational quantum well, providing bounds based on experimental data.

Contribution

It introduces a specific deformation of the Heisenberg algebra to model minimal length effects and compares its predictions with noncommutative geometry approaches and experimental results.

Findings

Perturbation analysis reveals distinct spectral signatures from different approaches.

The minimal length scale is constrained by GRANIT experiment data.

The upper bound on the minimal length is weaker than that from hydrogen atom studies.

Abstract

The dynamics of a particle in a gravitational quantum well is studied in the context of nonrelativistic quantum mechanics with a particular deformation of a two-dimensional Heisenberg algebra. This deformation yields a new short-distance structure characterized by a finite minimal uncertainty in position measurements, a feature it shares with noncommutative theories. We show that an analytical solution can be found in perturbation and we compare our results to those published recently, where noncommutative geometry at the quantum mechanical level was considered. We find that the perturbations of the gravitational quantum well spectrum in these two approaches have different signatures. We also compare our modified energy spectrum to the results obtained with the GRANIT experiment, where the effects of the Earth's gravitational field on quantum states of ultra cold neutrons moving above a mirror are studied. This comparison leads to an upper bound on the minimal length scale induced by the deformed algebra we use. This upper bound is weaker than the one obtained in the context of the hydrogen atom but could still be useful if the deformation parameter of the Heisenberg algebra is not a universal constant but a quantity that depends on the energetic content of the system.