Abstract
We obtain an exact one-dimensional time-dependent solution for a wave function ψ(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 |ψ(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.
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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 177, No. 3, pp. 497–517, December, 2013.
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Los, V.F., Los, N.V. Exact solution of the one-dimensional time-dependent Schrödinger equation with a rectangular well/barrier potential and its applications. Theor Math Phys 177, 1706–1721 (2013). https://doi.org/10.1007/s11232-013-0128-8
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DOI: https://doi.org/10.1007/s11232-013-0128-8