  We apply the formalism of quantum cosmology to models containing a phantom
field. Three models are discussed explicitly: a toy model, a model with an
exponential phantom potential, and a model with phantom field accompanied by a
negative cosmological constant. In all these cases we calculate the classical
trajectories in configuration space and give solutions to the Wheeler-DeWitt
equation in quantum cosmology. In the cases of the toy model and the model with
exponential potential we are able to solve the Wheeler-DeWitt equation exactly.
For comparison, we also give the corresponding solutions for an ordinary scalar
field. We discuss in particular the behaviour of wave packets in
minisuperspace. For the phantom field these packets disperse in the region that
corresponds to the Big Rip singularity. This thus constitutes a genuine quantum
region at large scales, described by a regular solution of the Wheeler-DeWitt
equation. For the ordinary scalar field, the Big-Bang singularity is avoided.
Some remarks on the arrow of time in phantom models as well as on the relation
of phantom models to loop quantum cosmology are given.
