  Using the Gell-Mann and Hartle formalism of generalized quantum mechanics of
closed systems, we study coarse-grained decoherent histories. The system under
consideration is one-dimensional and consists of a particle coupled to a von
Neumann apparatus that measures its position. The particle moves in a quadratic
potential; in particular we consider a driven harmonic oscillator. The real
line is divided into intervals of the same length, and coarse-grained histories
are defined by the arithmetic average of the initial and final position of the
particle to be within one of these intervals. The position of the pointer
correlates with this arithmetic average. In addition a constant term is added
to this average, which is a result of the presence of a driving force on the
oscillator. We investigate decoherence for such histories via the decoherence
functional for the particle-apparatus. An exact expression for this functional
has been derived. If the particle or the pointer of the apparatus is in an
exact position state initially, then exact decoherence ensues, and the relative
probability for such histories is obtained. If the initial state of the
particle, which is centered at some arbitrary point in the $x$-axis, is narrow
compared to the width of the intervals on the $x$-axis, then we obtain
approximate decoherence. The same result follows if the initial state of the
pointer is narrow. In these two situations, we evaluate expressions for the
decoherence functional qualitatively and quantitatively. In the latter case we
obtain the first two terms in an expansion in powers of the width of the
initial state of the particle or the pointer respectively. Approximate relative
probabilities can be assigned in this case, and we have obtained an expansion
up to second order in powers of the width of the initial state for an
expression for an upper bound of these probabilities.
