# Two-dimensional Noncommutative Gravitational Quantum Well

## TL;DR

This paper investigates two-dimensional noncommutative space-time effects on gravitational quantum wells, deriving bounds on noncommutativity parameters from neutron experiments and highlighting minimal length scales.

## Contribution

It introduces a framework for analyzing two-dimensional noncommutative gravitational quantum wells and estimates upper bounds on noncommutativity parameters from experimental data.

## Key findings

- Minimal length is smaller than a few nanometers.
- Upper bounds on noncommutativity parameters are established.
- Experimental results constrain noncommutative space-time models.

## Abstract

In this paper we consider two kinds of noncommutative space-time commutation relations in two-dimensional configuration space and feature the absolute value of the minimal length from the generalized uncertainty relations associated to the particular commutation relations. We study the problem of the two-dimensional gravitational quantum well in the new Hermitian variables and confront the experimental results for the first lowest energy state of the neutrons in the Earth's gravitational field to estimate the upper bounds on the noncommutativity parameters. The absolute value of the minimum length is smaller than a few nanometers.