  We probe the low-temperature behavior of a system of quantum bouncers as a
theoretical model for ultracold neutrons within a low energy modified version
of the standard quantum mechanics, due to the gravitational effects. Working in
one dimension, the energy spectrum and bound states of a deformed quantum
bouncer are obtained using the first-order WKB approximation, granted the very
low energy regime of the particle. In this manner, we can study energy levels
of a system of ultracold neutrons as an informative probe towards exploring the
low energy manifestation of semi-classical quantum gravitational effects. Our
calculated energy levels of ultracold neutrons are in accordance with the
observed energy levels, as obtained in the famous Nesvizhevsky \emph{et al.}
experiment, with a negative constant deformation, as dependent on the
deformation parameter. In advance, we tackle modified thermodynamics of a
system of quantum bouncers in the infrared regime via an ensemble theory both
in one dimension and also three dimensions, to seek for any trace of an
effective, thermodynamic dimensional reduction in this low energy regime of
semi-classical quantum gravity. While the issue of dimensional reduction has
been essentially assigned to the high energy regime, here we show that there is
a trace of an effective, thermodynamic dimensional reduction in infrared regime
with one important difference: in the high energy regime, the dimensional
reduction effectively occurs from $D=3$ to $D=1$, but here, in this low energy
regime, there is a trace of thermodynamic dimensional reduction from $D=3$ to
$D=2$.
