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arXiv:1507.03341 · Physica Scripta

Scattering by a reflectionless modified Pöschl–Teller potential: Bohmian trajectories and arrival times

S. V. Mousavi

Published Sep 1, 2015• 2 citations
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Abstract

A nonreflecting wavepacket is constructed by the superposition of reflectionless eigenstates of sech2 potential. Free propagation and propagation in the presence of the above potential of such a wavepacket is considered using the concept of arrival time. Comparison is made with the free evolving Gaussian wavepacket. Mean arrival time at a detector behind the well is given as a function of mass for separate cases. A selection of Bohmian trajectories in the interacting case are computed and compared to the trajectories of a free particle guided by a Gaussian packet.

DOI
10.1088/0031-8949/90/9/095001

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A nonreflecting wavepacket is constructed by the superposition of reflectionless eigenstates of sech2 potential. Free propagation and propagation in the presence of the above potential of such a wavepacket is considered using the concept of arrival time. Comparison is made with the free evolving Gaussian wavepacket. Mean arrival time at a detector behind the well is given as a function of mass for separate cases. A selection of Bohmian trajectories in the interacting case are computed and compared to the trajectories of a free particle guided by a Gaussian packet.
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We study the scattering of a Gaussian wave packet in the modified Pöschl–Teller potential well in time respect. Analytical expressions of average dwell time, phase time, and average traversal time due to Larmor-clock times are obtained, from which the average traversal time can be identified with the average dwell time. We observe that the average dwell time is appreciably longer than the phase time at low energies, but they tend to coincide in the limit of narrow momentum distribution, high energy, and large interaction region. We also find that both of them are longer than the classical scattering time in the potential. All of these times are, however, shorter than the free-particle time, demonstrating the acceleration effect of the potential in temporal aspect of the scattering.
Abstract We investigate the scattering of a wave packet by the Pöschl-Teller potential in momentum representation. The scattering dynamics of the wave packet for a long-time evolution is feasible in this representation. With the wave function in momentum space, we can construct the time-dependent phase space Wigner function. The corresponding density function in coordinate space is then calculated through the Wigner function. The reflectionless wave packet for integer ν and partially reflected for non-integer ν are demonstrated by analyzing the Wigner function.
The specific features of scattering of the Gaussian wave packet by the simplest of all reflectionless potentials, the Sech-squared well, are considered. The results of numerical computation show that a wave packet that has passed through this potential overtakes a free wave packet and is both higher and narrower than a free wave packet; in other words, the scattered wave packet accelerates inside the well and preserves its shape better. Comparisons with scattering by the square well and by a series of reflectionless potentials are also presented. Possible applications of the Sech-squared as well as other reflectionless potentials are discussed.
Abstract Bohmian trajectories are considered for a particle that is free (i.e. the potential energy is zero), except for a half-line barrier. On the barrier, both Dirichlet and Neumann boundary conditions are considered. The half-line barrier yields one of the simplest cases of diffraction. Using the exact time-dependent propagator found by Schulman, the trajectories are computed numerically for different initial Gaussian wave packets. In particular, it is found that different boundary conditions may lead to qualitatively different sets of trajectories. In the Dirichlet case, the particles tend to be more strongly repelled. The case of an incoming plane wave is also considered. The corresponding Bohmian trajectories are compared with the trajectories of an oil drop hopping on the surface of a vibrating bath.
Abstract How to compute the probability distribution of a detection time, i.e., of the time which a detector registers as the arrival time of a quantum particle, is a long-debated problem. In this regard, Bohmian mechanics provides in a straightforward way the distribution of the time at which the particle actually does arrive at a given surface in 3-space in the absence of detectors. However, as we discuss here, since the presence of detectors can change the evolution of the wave function and thus the particle trajectories, it cannot be taken for granted that the arrival time of the Bohmian trajectories in the absence of detectors agrees with the one in the presence of detectors, and even less with the detection time. In particular, we explain why certain distributions that Das and Dürr (Sci. Rep. 9: 2242, 2019) presented as the distribution of the detection time in a case with spin, based on assuming that all three times mentioned coincide, are actually not what Bohmian mechanics predicts.