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We investigate a cosmological model whose energy content is described by aChaplygin gas represented by a scalar field $\phi$ with an associated potentialproducing a big bang singularity such that for vanishing scale factor, $a\to0$, one has $|\phi|\to \infty$. The classical version of the model is discussedin detail, however, our main interest lies in its quantization. Uponquantization of this model in the Schr\"odinger picture, we get theWheeler--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 wavefunction should vanish at the classical singularity in order to avoid the bigbang, we show that a solution to the Wheeler-DeWitt equation fulfilling thiscondition can indeed be found. In addition to DeWitt's initial condition at thebig bang, we postulate an asymptotic condition to be imposed on the wavefunction which guarantees that the quantum wave function is strongly peaked atthe classical field configurations in the semiclassical limit. We alsoinvestigate a universe filled with dust (describing baryonic and dark matter)and show that in this case there exists an exact solution to the Wheeler-DeWittequation which describes the evolution of the universe during its wholehistory. Also for this model one can construct a wave packet which avoids thebig bang singularity at small scale factors and is strongly peaked at theclassical field in the far future of the universe. Finally, we discuss thesignificance of our treatment of the Wheeler-DeWitt equation and mention someopen problems.

Quantum Cosmology Close to the Classical Big Bang Singularity and in the Semiclassical Limit