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An-Najah University Journal for Research - A (Natural Sciences)

Strongly Singular Potentials in One Dimension

Sami M. Al-Jaber

Published Nov 1, 2005• 2 citations
Publisher

Abstract

Strongly singular potentials in one dimension are analyzed, namely the n th derivative interaction     x n  .For the repulsive case, the reflection and transmission coefficients are derived.It is shown that for n = even, these coefficients satisfy the unitarity of the scattering matrix for all values of the incident energy and the barrier becomes perfectly reflective for a specific value of the energy.For n=odd 1  , however, the coefficients do not satisfy the unitarity of the scattering matrix except for one specific value of the incident energy at which the barrier becomes perfectly reflective.For the n=1 case, the results showed that the coefficients satisfy the unitarity of the scattering matrix.For the attractive case, both the bound states and the scattering states are examined.It is shown that, for n=even 0  , there exist two bound states and only one bound state for n=0.For the scattering states, it is demonstrated that one recovers the same coefficients that were obtained for the repulsive case.

DOI
10.35552/anujr.a.19.1.603

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Strongly singular potentials in one dimension are analyzed, namely the n th derivative interaction     x n  .For the repulsive case, the reflection and transmission coefficients are derived.It is shown that for n = even, these coefficients satisfy the unitarity of the scattering matrix for all values of the incident energy and the barrier becomes perfectly reflective for a specific value of the energy.For n=odd 1  , however, the coefficients do not satisfy the unitarity of the scattering matrix except for one specific value of the incident energy at which the barrier becomes perfectly reflective.For the n=1 case, the results showed that the coefficients satisfy the unitarity of the scattering matrix.For the attractive case, both the bound states and the scattering states are examined.It is shown that, for n=even 0  , there exist two bound states and only one bound state for n=0.For the scattering states, it is demonstrated that one recovers the same coefficients that were obtained for the repulsive case.
In this paper, the results of the analysis of the 1-dimensional potential barrier case are presented when the energy of the incident beam is equal to the potential of the system (𝐸 = 𝑉 0 ).The analysis was carried out using a computational simulation based on Python programming by reviewing the values of the reflection and transmission coefficients and their influence on the amplitude of the wave function.As an example case for simulation, the potential value of the system 𝑉 0 = 60 × 10 -39 J is taken.First, an approach is made when the system energy E approaches the potential 𝑉 0 from below (𝐸 ≲ 𝑉 0 ), then an approach is made when the system energy E approaches the potential 𝑉 0 from above (𝐸 ≳ 𝑉 0 ).From these two approaches, the result is that for both the 𝐸 ≲ 𝑉 0 and 𝐸 ≳ 𝑉 0 cases, the graph shape is almost the same with slightly different amplitudes.The amplitude for the system with 𝐸 = 61 × 10 -39 J is larger than for the system 𝐸 = 59 × 10 -39 J.An important event is that when the energy of the incident beam is equal to the potential of the system (𝐸 = 𝑉 0 ), the reflection and transmission coefficient values are undefined so that no amplitude appears in the visualization of the programming results.Thus, it is impossible to obtain information about the particle beam in the potential region 𝑉 0 .
Transmission and scattering problems involving complex potentials are important in physics, in particular in describing nuclear collisions. We describe many pedagogical features of transmission in one dimension and scattering and absorption cross sections in three dimensions for a rectangular absorptive potential and compare the results with scattering from real barriers and a hard sphere. For a given energy in one dimension we show that the absorption reaches a maximum for a critical value of the absorption strength and then monotonically decreases with an increase in the absorption strength. An infinitely absorptive well becomes almost fully reflective and thus is similar to an infinitely high barrier. Similar results are found in three dimensions. We show that the absorption cross section in three dimensions diverges at a threshold energy, which is of critical importance for exothermic reactions.
The scattering of quantum particles by non-Hermitian (generally non-local) potentials in one dimension may result in asymmetric transmission and/or reflection from left and right incidence. After extending the concept of symmetry for non-Hermitian potentials, eight generalized symmetries based on the discrete Klein's four-group (formed by parity, time reversal, their product, and unity) are found. Together with generalized unitarity relations they determine selection rules for the possible and/or forbidden scattering asymmetries. Six basic device types are identified when the scattering coefficients (squared moduli of scattering amplitudes) adopt zero/one values, and transmission and/or reflection are asymmetric. They can pictorically be described as a one-way mirror, a one-way barrier (a Maxwell pressure demon), one-way (transmission or reflection) filters, a mirror with unidirectional transmission, and a transparent, one-way reflector. We design potentials for these devices and also demonstrate that the behavior of the scattering coefficients can be extended to a broad range of incident momenta.
The quantum mechanical problem of a particle moving in one dimension (1D) under the influence of a singular potential is reexamined. An attractive delta function potential of infinite strength and the 1D Coulomb potential is considered in particular. In both cases the singularity acts as an impenetrable barrier, thus dividing space into two independent regions. Analysis of the scattering states gives results in agreement with this conclusion. The implications that the bound states of the 1D Coulomb potential cannot be classified according to parity, even though the potential is symmetric, is also discussed herein.