  An exact time-dependent solution for the wave function $\psi(r,t)$ of a
particle moving in the presence of an asymmetric rectangular well/barrier
potential varying in one dimension is obtained by applying a novel for this
problem approach using multiple scattering theory (MST) for the calculation of
the space-time propagator. This approach, based on the localized at the
potential jumps effective potentials responsible for transmission through and
reflection from the considered rectangular potential, enables considering these
processes from a particle (rather than a wave) point of view. The solution
describes these quantum phenomena as a function of time and is related to the
fundamental issues (such as measuring time) of quantum mechanics. It is
presented in terms of integrals of elementary functions and is a sum of the
forward- and backward-moving components of the wave packet. The relative
contribution of these components and their interference as well as of the
potential asymmetry to the probability density $|\psi(x,t)|^2$ and particle
dwell time is considered and numerically visualized for narrow and broad energy
(momentum) distributions of the initial Gaussian wave packet. The obtained
solution is also related to the kinetic theory of nanostructures due to the
fact that the considered potential can model the spin-dependent potential
profile of the magnetic multilayers used in spintronics devices.
