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Physica Scripta

Scattering of a wave packet by the Pöschl-Teller potential well

Tsin-Fu Jiang

Published Nov 23, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ad0f7c

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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.
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 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.
A mapping that relates the Wigner phase-space distribution function of a given stationary quantum mechanical wave function, a solution of the Schr\"odinger equation, to a specific solution of the Liouville equation, both subject to the same potential, is studied. By making this mapping, bound states are described by semiclassical distribution functions still depending on Planck's constant, whereas for elastic scattering of a particle by a potential they do not depend on it, the classical limit being reached in this case. Following this method, the mapped distributions of a particle bound in the P\"oschl-Teller potential and also in a modified oscillator potential are obtained.
In the present paper the quantum mechanical representation in phase space introduced by Torres-Vega and Frederick (1993 J. Chem. Phys. 98 3103) has been successfully applied to the phase space description of the quantum potential scattering. 3D potential scattering is formulated from the view-points of the wave function and density operator in this new quantum phase-space representation. It is found that in the region where the potential acts predominantly, the scattered part of the wave function or density in quantum phase space depends closely not only on the potential, but also on a non-locality factor arising from the use of the eigenfunctions of the coordinate and momentum operators in the phase-space representation. Transition from the present formulation to the classical scattering is made.
Wave-packet scattering from a stationary potential is significantly modified when the wave packet is subject to an external time-dependent force during the interaction. In the semiclassical limit, wave-packet motion is simply described by Newtonian equations, and the external force can, for example, cancel the potential force, making a potential barrier transparent. Here we consider wave-packet scattering from reflectionless potentials, where in general the potential becomes reflective when probed by an accelerated wave packet. In the particular case of the recently introduced class of complex Kramers-Kronig potentials we show that a broad class of time-dependent forces can be applied without inducing any scattering, while there is a breakdown of the reflectionless property when there is a broadband distribution of initial particle momentum, involving both positive and negative components.