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arXiv:quant-ph/0504128

Wath Time an Incident Wave Packet is Reflected?

Kh. Saaidi

Published Apr 17, 2005• 0 citations
arXivHTMLPDF

Abstract

We study the one dimensional time dependent Schrödinger equation for a potential step with $E < V_0$. We obtain the wave is not instantaneously reflected, but for a few moment of time the wave packet penetrate into inaccessible classically region and then reflected. So it is shown that one can write the peneterated wave paket as superposition of incident and reflected wave packets like.

DOI
10.48550/arxiv.quant-ph/0504128

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We study the one dimensional time dependent Schrödinger equation for a potential step with E<V0E < V_0E<V0​. We obtain the wave is not instantaneously reflected, but for a few moment of time the wave packet penetrate into inaccessible classically region and then reflected. So it is shown that one can write the peneterated wave paket as superposition of incident and reflected wave packets like.
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We investigate the transmission and reflection of a quantum particle incident upon a step potential decrease and a square well. The probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation are in excellent agreement. We explain why the probabilities agree so well. In doing so, we make use of an exact analytical expression for the square well for time-dependent transmission and reflection, which reveals additional interesting and unexpected results. One such result is that transmission of a wave packet can occur with the probability of transmission depending weakly on the initial spread of the packet. The explanations and the additional results will be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.
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We investigate the transmission and reflection of a quantum particle incident upon a step potential increase, a step potential decrease, a square well, and a square barrier, all well studied in undergraduate quantum mechanics. We are especially interested in the extreme where the change in the potential is arbitrarily large, but with the difference in the energy of the particle and the potential held fixed, if possible. We obtain the probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation. In the time-dependent case, we have the particle initially in a Gaussian wave packet or a similar localized state. We obtain results that fall into three categories: results that are not surprising, results where time-dependent and time-independent agree surprisingly well, and results that are very different. We discuss the unexpected results. Our work may be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.