  We propose and analyze a scanning microscope to monitor `live' the quantum
dynamics of cold atoms in a Cavity QED setup. The microscope measures the
atomic density with subwavelength resolution via dispersive couplings to a
cavity and homodyne detection within the framework of continuous measurement
theory. We analyze two modes of operation. First, for a fixed focal point the
microscope records the wave packet dynamics of atoms with time resolution set
by the cavity lifetime. Second, a spatial scan of the microscope acts to map
out the spatial density of stationary quantum states. Remarkably, in the latter
case, for a good cavity limit, the microscope becomes an effective quantum
non-demolition (QND) device, such that the spatial distribution of motional
eigenstates can be measured back-action free in single scans, as an emergent
QND measurement.
