  We define the current of a quantum observable and, under well defined
conditions, we connect its ensemble average to the index of a Fredholm
operator. The present work builds on a formalism developed by Kellendonk and
Schulz-Baldes \cite{Kellendonk:2004p597} to study the quantization of edge
currents for continuous magnetic Schroedinger operators. The generalization
given here may be a useful tool to scientists looking for novel manifestations
of the topological quantization. As a new application, we show that the
differential conductance of atomic wires is given by the index of a certain
operator. We also comment on how the formalism can be used to probe the
existence of edge states.
