  The not necessarily unitary evolution operator of a finite dimensional
quantum system is studied with the help of a projection operators technique.
Applying this approach to the Schr\"odinger equation allows the derivation of
an alternative expression for the evolution operator, which differs from the
traditional chronological exponent. An appropriate choice of projection
operators results in the possibility of studying the diagonal and non-diagonal
elements of the evolution operator separately. The suggested expression implies
a particular form of perturbation expansion, which leads to a new formula for
the short time dynamics. The new kind of perturbation expansion can be used to
improve the accuracy of the usual chronological exponent significantly. The
evolution operator for any arbitrary time can be efficiently recovered using
the semigroup properties. The method is illustrated by two examples, namely the
dynamics of a three-level system in two nonresonant laser fields and the
calculation of the partition function of a finite XY-spin chain.
