Geodesic
Exact solution of the~one-dimensional time-dependent Schr\"odinger equation with a~rectangular well/barrier potential and its applications
Teoretičeskaâ i matematičeskaâ fizika, Tome 177 (2013) no. 3, pp. 497-517

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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.
Keywords: time-dependent Schrödinger equation, rectangular well/barrier potential, backward-moving wave, dwell time, time of arrival. 
@article{TMF_2013_177_3_a5,
     author = {V. F. Los and N. V. Los},
     title = {Exact solution of the~one-dimensional time-dependent {Schr\"odinger} equation with a~rectangular well/barrier potential and its applications},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {497--517},
     publisher = {mathdoc},
     volume = {177},
     number = {3},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2013_177_3_a5/}
}
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V. F. Los; N. V. Los. Exact solution of the~one-dimensional time-dependent Schr\"odinger equation with a~rectangular well/barrier potential and its applications. Teoretičeskaâ i matematičeskaâ fizika, Tome 177 (2013) no. 3, pp. 497-517. http://geodesic.mathdoc.fr/item/TMF_2013_177_3_a5/