  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.
