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Modern Physics Letters B

SCATTERING TIMES IN THE MODIFIED PÖSCHL–TELLER POTENTIAL WELL

Chang-Soo Park

Published Mar 14, 2012• 2 citations
Publisher

Abstract

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.

DOI
10.1142/s0217984912500467

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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.
A {\it completed} scattering of a particle on a static one-dimensional (1D) potential barrier is a combined quantum process to consist from two elementary sub-processes (transmission and reflection) evolved coherently at all stages of scattering and macroscopically distinct at the final stage. The existing model of the process is clearly inadequate to its nature: all one-particle "observables" and "tunneling times", introduced as quantities to be common for the sub-processes, cannot be experimentally measured and, consequently, have no physical meaning; on the contrary, quantities introduced for either sub-process have no basis, for the time evolution of either sub-process is unknown in this model. We show that the wave function to describe a completed scattering can be uniquely presented as the sum of two solutions to the Schrödinger equation, which describe separately the sub-processes at all stages of scattering. For symmetric potential barriers such solutions are found explicitly. For either sub-process we define the time spent, on the average, by a particle in the barrier region. We define it as the Larmor time. As it turned out, this time is just Buttiker's dwell time averaged over the corresponding localized state. Thus, firstly, we justify the known definition of the local dwell time introduced by Hauge and co-workers as well by Leavens and Aers, for now this time can be measured; secondly, we confirm that namely Buttiker's dwell time gives the energy-distribution for the tunneling time; thirdly, we state that all the definitions are valid only if they are based on the wave functions for transmission and reflection found in our paper. Besides, we define the exact and asymptotic group times to be auxiliary in timing the scattering process.
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.
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.
In this paper we compare the concept of the tunneling time introduced in quant-ph/0405028 with those of the phase and dwell times. As is shown, unlike the latter our definition of the transmission time coincides, in the limit of weak scattering potentials, with that for a free particle. This is valid for all values of the particle's momentum, including the case of however slow particles. All three times are also considered for a resonant tunneling. In all the cases the main feature to distinguish our concept from others is that the average starting point of transmitted (reflected) particles does not coincide with that of all particles. One has to stress here that there is no such an experiment which would give coordinates of all the three points, simultaneously. For measuring the position of the average starting point of transmitted particles we propose an experimental scheme based on the Larmor precession effect.
We consider the quantum traversal time of an incident wave packet across a potential well using the theory of quantum time of arrival (TOA) operators. This is done by constructing the corresponding TOA operator across a potential well via quantization. The expectation value of the potential-well TOA operator is compared to the free-particle case for the same incident wave packet. The comparison yields a closed-form expression of the quantum well traversal time which explicitly shows the classical contributions of the positive and negative momentum components of the incident wave packet and a purely quantum-mechanical contribution significantly dependent on the well depth. An incident Gaussian wave packet is then used as an example. It is shown that for shallow potential wells, the quantum well traversal time approaches the classical traversal time across the well region when the incident wave packet is spatially broad and approaches the expected quantum free-particle traversal time when the wave packet is localized. For deep potential wells, the quantum traversal time oscillates from positive to negative, implying that the wave packet can be advanced or delayed.