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Chemical Physics Letters

Quantum wave packet dynamics with trajectories: reflections on a downhill ramp potential

Courtney L. Lopreore, Róbert E. Wyatt

Published Jul 1, 2000• 44 citations
Publisher
DOI
10.1016/s0009-2614(00)00620-5

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The transmission and the reflection of a wave incident on a potential in one-dimensional quantum mechanics are usually discussed by assuming that a plane wave eikx or a broad wave packet is incident. We are interested in what happens if the width of the incident wave packet is very small. In this case, the incident wave packet can spread out in such a way that part of the wave packet turns around before reaching the potential area and proceeds backward. This can clearly be visualized by means of the Bohm trajectories associated with the wave packet. We discuss a complication that arises in this case regarding the notion of the reflection probability.PACS Nos.: 03.65–w, 03.65.Nk

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Journal of Mathematical Physics•May 1, 2019•1 citation
The quantum trajectory method (QTM) was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for N fluid elements (particles) moving under the influence of both the force from the potential surface and from the quantum potential. In this study, distributed approximating functionals (DAFs) are used on a uniform grid to compute the necessary derivatives in the equations of motion. Transformations between the physical grid where the particle coordinates are defined and the uniform grid are handled through a Jacobian, which is also computed using DAFs. A difficult problem associated with computing derivatives on finite grids is the edge problem. This is handled effectively by using DAFs within a least squares approach to extrapolate from the known function region into the neighboring regions. The QTM–DAF is then applied to wave packet transmission through a one-dimensional Eckart potential. Emphasis is placed upon computation of the transmitted density and wave function. A problem that develops when part of the wave packet reflects back into the reactant region is avoided in this study by introducing a potential ramp to sweep the reflected particles away from the barrier region.
The transmission and the reflection of a wave incident on a potential in one-dimensional quantum mechanics are usually discussed by assuming that a plane wave eikx or a broad wave packet is incident. We are interested in what happens if the width of the incident wave packet is very small. In this case, the incident wave packet can spread out in such a way that part of the wave packet turns around before reaching the potential area and proceeds backward. This can clearly be visualized by means of the Bohm trajectories associated with the wave packet. We discuss a complication that arises in this case regarding the notion of the reflection probability.PACS Nos.: 03.65–w, 03.65.Nk
We complete our study on the uncertainties in position and momentum associated with the semiclassical Hagedorn wave packets by first filling in a technical gap in the dynamics of bound states for isochronous potentials. We then consider scattered states and show that, if the packet is reflected from the potential or transmitted through a symmetric potential, then a minimal uncertainty “initial” state cannot in general lead to a “final” state with minimal uncertainty, and we give an explicit relationship for the difference in terms of a characteristic time associated with classical trajectories. We also characterize the behavior of the uncertainty product in the case where the underlying classical dynamics lead to capture by the potential.