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An exact time-dependent solution for the wave function $\psi(r,t)$ of aparticle moving in the presence of an asymmetric rectangular well/barrierpotential varying in one dimension is obtained by applying a novel for thisproblem approach using multiple scattering theory (MST) for the calculation ofthe space-time propagator. This approach, based on the localized at thepotential jumps effective potentials responsible for transmission through andreflection from the considered rectangular potential, enables considering theseprocesses from a particle (rather than a wave) point of view. The solutiondescribes these quantum phenomena as a function of time and is related to thefundamental issues (such as measuring time) of quantum mechanics. It ispresented in terms of integrals of elementary functions and is a sum of theforward- and backward-moving components of the wave packet. The relativecontribution of these components and their interference as well as of thepotential asymmetry to the probability density $|\psi(x,t)|^2$ and particledwell time is considered and numerically visualized for narrow and broad energy(momentum) distributions of the initial Gaussian wave packet. The obtainedsolution is also related to the kinetic theory of nanostructures due to thefact that the considered potential can model the spin-dependent potentialprofile of the magnetic multilayers used in spintronics devices.

A multiple scattering theory approach to solving the time-dependent Schr\"odinger equation with an asymmetric rectangular potential