  This paper systematically treats the evolving quantum state for
two-dimensional black holes, with particular focus on the CGHS model, but also
elucidating features generalizing to higher dimensions. This is done in
Schr\"odinger picture(s), to exhibit the dynamic evolution of the state at
intermediate times. After a review of classical solutions, also connecting to
descriptions of higher-dimensional black holes, it overviews the canonical
quantum treatment of the full evolution, including gravitational dynamics.
Derived in an approximation to this, following conversion to "perturbation
picture," is the evolution of the quantum matter on the background geometry.
Features of the evolving matter state are described, based on choice of a time
slicing to put the evolution into ADM form. The choices of slicing as well as
coordinates on the slices result in different quantum "pictures" for treating
the evolution. If such a description is based on smooth trans-horizon slices,
that avoids explicit reference to ultra-planckian modes familiar from
traditional treatments, and exhibits the Hawking excitations as emerging from a
"quantum atmosphere" with thickness comparable to the inverse temperature.
Detailed study of the state exhibits the entanglement structure between Hawking
quanta and the partner excitations inside the black hole, and the corresponding
"missing information." This explicit description also allows direct study of
the evolution and features, e.g. as seen by infalling observers, of these
partner excitations, helping to address various puzzles with them. Explicit
treatment of the evolving state, and its extension to higher dimensions,
provides further connections to information theory and a starting point for
study of corrections that can unitarize evolution, arising from new quantum
gravity effects -- whether wormholes or something entirely different.
