  The time evolution of a quantum wave packet in the linear gravity potential
is known as Quantum Bouncing Ball. The qBounce collaboration recently observed
such a system by dropping wave packets of ultracold neutrons by a height of
roughly 30 microns. In this article, space and momentum spectra as well as
Wigner functions of the neutron wave functions in the gravitational field of
the Earth are analyzed. We investigate the quantum states in the "preparation
region", into which they transition after exiting a narrow double-mirror system
and where we would expect to observe free fall and bounces in classical
physics. For this, we start from the stationary solutions and eigenvalues of
the Schr\"odinger equation in terms of Airy functions and their zeros.
Subsequently, we examine space and momentum distributions as well as Wigner
functions in phase space for pure and mixed quantum states. The eventual
influence of Yukawa-like forces for small distances of several micrometers from
the mirror is included through first order perturbation calculations. Those
allow us to study the resulting modifications of space and momentum
distributions, and phase space functions.
