  We examine several numerical techniques for the calculation of the dynamics
of quantum systems. In particular, we single out an iterative method which is
based on expanding the time evolution operator into a finite series of
Chebyshev polynomials. The Chebyshev approach benefits from two advantages over
the standard time-integration Crank-Nicholson scheme: speedup and efficiency.
Potential competitors are semiclassical methods such as the Wigner-Moyal or
quantum tomographic approaches. We outline the basic concepts of these
techniques and benchmark their performance against the Chebyshev approach by
monitoring the time evolution of a Gaussian wave packet in restricted
one-dimensional (1D) geometries. Thereby the focus is on tunnelling processes
and the motion in anharmonic potentials. Finally we apply the prominent
Chebyshev technique to two highly non-trivial problems of current interest: (i)
the injection of a particle in a disordered 2D graphene nanoribbon and (ii) the
spatiotemporal evolution of polaron states in finite quantum systems. Here,
depending on the disorder/electron-phonon coupling strength and the device
dimensions, we observe transmission or localisation of the matter wave.
