  This paper explores the generalization of the concept of a "probability
current", familiar from wave-function quantum mechanics, to quantum systems
with finite-dimensional Hilbert spaces. The generalized definition applies both
to isolated systems evolving via the Schr\"odinger equation and to more general
open systems obeying the Lindblad master equation. We establish several
properties of the probability current and explore its relation to thermodynamic
heat and work.
