  A Schr\"odinger-picture description of the evolving quantum state of Hawking
radiation is given, based on an ADM decomposition using time slicings that
smoothly cross the horizon. This treatment avoids requiring a role for
trans-planckian modes, which can be viewed as artifacts of Hawking's original
calculation, and also supports arguments that radiation from black holes is
produced in a "quantum atmosphere" with thickness comparable to the horizon
size, rather than microscopically far from it. Particularly explicit formulas
are given for the two-dimensional analog of the Schwarzschild geometry. This
analysis is expected to generalize to other black holes, and to cosmology. The
resulting quantum evolution also provides important background for
investigating corrections to the Hawking process, as are necessary for
restoring unitary evolution of black hole decay.
