Two-dimensional noncommutative gravitational quantum well

Abstract

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