  We discuss the numerical solution of the Schr\"odinger equation with a
time-dependent Hamilton operator using commutator-free time-propagators. These
propagators are constructed as products of exponentials of simple weighted sums
of the Hamilton operator. Owing to their exponential form they strictly
preserve the unitarity of time-propagation. The absence of commutators or other
computationally involved operations allows for straightforward implementation
and application also to large scale and sparse matrix problems. We explain the
derivation of commutator-free exponential time-propagators in the context of
the Magnus expansion, and provide optimized propagators up to order eight. An
extensive theoretical error analysis is presented together with practical
efficiency tests for different problems. Issues of practical implementation, in
particular the use of the Krylov technique for the calculation of exponentials,
are discussed. We demonstrate for two advanced examples, the hydrogen atom in
an electric field and pumped systems of multiple interacting two-level systems
or spins that this approach enables fast and accurate computations.
