  A simple quantum mechanical model of $N$ free scalar fields interacting with
a dynamical moving mirror is formulated and shown to be equivalent to
two-dimensional dilaton gravity. We derive the semi-classical dynamics of this
system, by including the back reaction due to the quantum radiation. We develop
a hamiltonian formalism that describes the time evolution as seen by an
asymptotic observer, and write a scattering equation that relates the
in-falling and out-going modes at low energies. At higher incoming energy flux,
however, the classical matter-mirror dynamics becomes unstable and the mirror
runs off to infinity. This instability provides a useful paradigm for black
hole formation and introduces an analogous information paradox. Finally, we
propose a new possible mechanism for restoring the stability in the
super-critical situation, while preserving quantum coherence. This mechanism is
based on the notion of an effective time evolution, that takes into account the
quantum mechanical effect of the measurement of the Hawking radiation on the
state of the infalling matter.
