  Evaluating a functional integral exactly over a subset of metrics that
represent the quantum fluctuations of the horizon of a black hole, we obtain a
Schroedinger equation in null coordinate time for the key component of the
metric. The equation yields a current that preserves probability if we use the
most natural choice of functional measure. This establishes the existence of
blurred horizons and a thermal atmosphere. It has been argued previously that
the existence of a thermal atmosphere is a direct concomitant of the thermal
radiation of black holes when the temperature of the hole is greater than that
of its larger environment, which we take as zero.
