  We investigate a cosmological model whose energy content is described by a
Chaplygin gas represented by a scalar field $\phi$ with an associated potential
producing a big bang singularity such that for vanishing scale factor, $a\to
0$, one has $|\phi|\to \infty$. The classical version of the model is discussed
in detail, however, our main interest lies in its quantization. Upon
quantization of this model in the Schr\"odinger picture, we get the
Wheeler--DeWitt equation which can be solved exactly in the two limits $a\to 0$
and $a\to\infty$, respectively. Employing the DeWitt criterium that the wave
function should vanish at the classical singularity in order to avoid the big
bang, we show that a solution to the Wheeler-DeWitt equation fulfilling this
condition can indeed be found. In addition to DeWitt's initial condition at the
big bang, we postulate an asymptotic condition to be imposed on the wave
function which guarantees that the quantum wave function is strongly peaked at
the classical field configurations in the semiclassical limit. We also
investigate a universe filled with dust (describing baryonic and dark matter)
and show that in this case there exists an exact solution to the Wheeler-DeWitt
equation which describes the evolution of the universe during its whole
history. Also for this model one can construct a wave packet which avoids the
big bang singularity at small scale factors and is strongly peaked at the
classical field in the far future of the universe. Finally, we discuss the
significance of our treatment of the Wheeler-DeWitt equation and mention some
open problems.
