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A propagator for the one dimensional time-dependent Schr\"odinger equationwith an asymmetric rectangular potential is obtained using themultiple-scattering theory approach. It allows for the consideration of thereflection and transmission processes as the particle scattering at thepotential jump (in contrast to the conventional wave-like picture) and foraccounting for the nonclassical counterintuitive contribution of thebackward-moving component of the wave packet attributed to the particle. Thispropagator completely resolves the corresponding time-dependent Schr\"odingerequation (defines the wave function ${\psi}(x,t)$) and allows for consideringthe quantum mechanical effects of a particle reflection from the potentialdownward step/well and a particle tunneling through the potential barrier as afunction of time. These results are related to fundamental issues such asmeasuring time in quantum mechanics (tunneling time, time of arrival, dwelltime). For imaginary time, which represents an inverse temperature$(t\rightarrow-i{\hbar}/k_{B}T)$, the obtained propagator is equivalent to thedensity matrix for a particle that is in a heat bath and is subject to arectangular potential. If one shifts to imaginary time $(t\rightarrow-it)$, thematrix element of the calculated propagator in the spatial basis provides asolution to the diffusion-like equation with a rectangular potential. Theobtained exact results are presented as the integrals from elementary functionsand thus allow for a numerical visualization of the probability density$|\psi(x,t)|^2$, the density matrix and the solution of the diffusion-likeequation. The results obtained may also be useful for spintronics applicationsdue to the fact that the asymmetric (spin-dependent) rectangular potential canmodel the potential profile in layered magnetic nanostructures.

An exact solution of the time-dependent Schr\"odinger equation with a rectangular potential for real and imaginary time