  We give a general description of the evolving quantum state of a
Schwarzschild black hole, in the quantum field theory approximation. Such a
time-dependent description is based on introducing a choice of time slices. We
in particular consider slices that smoothly cross the horizon, and introduction
of "stationary" such slices simplifies the analysis. This analysis goes beyond
standard derivations of Hawking radiation that focus on asymptotic excitations,
and in particular gives an evolving state that is regular at the horizon, with
no explicit transplanckian dependence, and that can in principle be generalized
to incorporate interacting fields. It is also expected to be useful in
connecting to information-theoretic investigation of black hole evolution. The
description of the evolving state depends on the choice of slices as well as
coordinates on the slices and mode bases; these choices give different
"pictures" analogous to that of Schr\"odinger. Evolution does have a simpler
appearance in an energy eigenbasis, but such a basis is also singular at the
horizon; evolution of regular modes has a more complicated appearance, whose
properties may be inferred by comparing with the energy eigenbasis. In a
regular description, Hawking quanta are produced in a black hole atmosphere, at
scales comparable to the horizon size. This approach is also argued to extend
to more general asymptotics, such as that of anti de Sitter space. In the
latter context, this analysis provides a description of the hamiltonian and
evolution of a black hole that may be compared to the large-$N$ dynamics of the
proposed dual CFT.
