  Following the evolution of an open quantum system requires full knowledge of
its dynamics. In this paper we consider open quantum systems for which the
Hamiltonian is ``uncertain''. In particular, we treat in detail a simple system
similar to that considered by Mabuchi [Quant. Semiclass. Opt. 8, 1103 (1996)]:
a radiatively damped atom driven by an unknown Rabi frequency $\Omega$ (as
would occur for an atom at an unknown point in a standing light wave). By
measuring the environment of the system, knowledge about the system state, and
about the uncertain dynamical parameter, can be acquired. We find that these
two sorts of knowledge acquisition (quantified by the posterior distribution
for $\Omega$, and the conditional purity of the system, respectively) are quite
distinct processes, which are not strongly correlated. Also, the quality and
quantity of knowledge gain depend strongly on the type of monitoring scheme. We
compare five different detection schemes (direct, adaptive, homodyne of the $x$
quadrature, homodyne of the $y$ quadrature, and heterodyne) using four
different measures of the knowledge gain (Shannon information about $\Omega$,
variance in $\Omega$, long-time system purity, and short-time system purity).
