Exact solution of the one-dimensional time-dependent Schrödinger equation with a rectangular well/barrier potential and its applications

We obtain an exact one-dimensional time-dependent solution for a wave function $\psi(x,t)$ of a particle moving in the presence of a rectangular well or barrier. We present the solution, which holds for both the well and the barrier, in terms of the integrals of elementary functions; it is the sum of forward- and backward-moving components of the wave packet. We consider and numerically visualize the relative contribution of these components and of their interference to the probability density $|\psi(x,t)|^{2}$ and the particle arrival time and dwell time for the narrow and broad energy (momentum) distributions of the initial Gaussian wave packet. We show that in the case of a broad initial wave packet, the quantum mechanical counterintuitive effect of the influence of the backward-moving components on the considered quantities becomes essential.