  Bianchi -I, -III, and FRW type models minimally coupled to a massive
spatially homogeneous scalar field (i.e. a particle) are studied in the
framework of semiclassical quantum gravity. In a first step we discuss the
solutions of the corresponding equation for a Schr\"odinger particle
propagating on a classical background. The back reaction of the Schr\"odinger
particle on the classical metric is calculated by means of the Wigner function
and by means of the expectation value of the energy-momentum-tensor of the
field as a source. Both methods in general lead to different results.
