  In summer 1999 an experiment at ILL, Grenoble was conducted. So-called
ultra-cold neutrons (UCN) were trapped in the vertical direction between the
Fermi-potential of a smooth mirror below and the gravitational potential of the
earth above [Ne00, Ru00]. If quantum mechanics turns out to be a sufficiently
correct description of the phenomena in the regime of classical, weak
gravitation, one should observe the forming of quantized bound states in the
vertical direction above a mirror. Already in a simplified view, the data of
the experiment provides strong evidence for the existence of such
gravitationally bound quantized states. A successful quantum-mechanical
description would then provide a convincing argument, that the socalled first
quantization can be used for gravitation as an interaction potential, as this
is widely expected. Furthermore, looking at the characteristic length scales of
about 10 mikron of such bound states formed by UCN, one sees, that a complete
quantum mechanical description of this experiment additionally would enable one
to check for possible modifications of Newtonian gravitation on distance scales
being one order of magnitude below currently available tests [Ad00]. The work
presented here deals mainly with the development of a quantum mechanical
description of the experiment.
