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Acta Physica Polonica A

On the Transmission Coefficient for the Double δ'-Function Potential

I. Yanetka

Published Jun 1, 2000• 2 citations
Publisher

Abstract

The unbound-state solution of the Schrödinger equation is examined for the potential that is defined as the sum of two δ'-functions of non-equal strengths.The analytical expression for the transmission coemcient is derived from the solution.The transmission coemcient has an absolute maximum at the zero wave number.Further, the transmission coemcient is shown to exhibit relative maxima and minima.Moreover, it is proved that the transmission coemcient in its relative maxima is larger and in its relative minima is smaller than the transmission coemcient for the corresponding single δ' -functiοn potential.

DOI
10.12693/aphyspola.97.1053

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The unbound-state solution of the Schrödinger equation is examined for the potential that is defined as the sum of two δ'-functions of non-equal strengths.The analytical expression for the transmission coemcient is derived from the solution.The transmission coemcient has an absolute maximum at the zero wave number.Further, the transmission coemcient is shown to exhibit relative maxima and minima.Moreover, it is proved that the transmission coemcient in its relative maxima is larger and in its relative minima is smaller than the transmission coemcient for the corresponding single δ' -functiοn potential.
The solution of the one-dimensional Dirac equation is examined for the symmetrical double delta-function potential. A formula for the relativistic transmission coefficient is derived from the solution. The maximum condition as well as the minimum condition for the transmission coefficient is determined. The transmission coefficient is shown to be equal to unity in its maximum. In its minimum, it is smaller than the transmission coefficient for the single delta-function potential of the twofold strength. Further, some differences between the relativistic and non-relativistic transmission coefficient are presented. It is found that the wave numbers, at which the maxima and minima occur according to the non-relativistic treatment, are shifted by the relativistic treatment. Unlike the non-relativistic transmission coefficient, the relativistic transmission coefficient does not approach unity in the range of high energies. It oscillates between unity and a constant. The value of this constant depends on the strength of the delta-interaction. Thus, the relativistic corrections to the transmission coefficient for the symmetrical double delta-function potential are found to represent significant shifts in its characteristics.
The energy and parity eigenstates for quadrupolar, triple δ-function potential are analysed. Using the analytical solutions in specific domains, simple expressions are obtained for even- and odd-parity bound-state energies. The Heisenberg uncertainty product is observed to have a minimum for a specific strength of the potential. The scattering states have a transmission resonance for some energies, and the poles of the transmission coefficients correspond to bound-state energies. These features are of considerable pedagogical interest in the understanding of quantum-mechanical properties of bound and scattering states, for graduate and general physicists.
Abstract The reflection and transmission coefficients describe the behavior of the matter wave incident on a potential barrier. They can be expressed in terms of the probability with which the matter wave can be reflected or transmitted. The central equation accounting for the behavior of the matter wave is the Schrödinger equation. The Schrödinger equation is the second order partial differential equation. However, in a stationary state, the Schrödinger equation is reduced to the time independent Schrödinger equation. This time independent Schrödinger equation is the second order linear ordinary differential equation. Since the time independent Schrödinger equation is linear, superposition of any of the two solutions to the time independent Schrödinger equation is also a solution. In this paper, we focus on the superposition of various potentials. The reflection and transmission coefficients from the superposition of various potentials are obtained. A comparison between the exact coefficients and those obtained by the 2 × 2 transfer matrix is made. The relationship between the transmission coefficient of the superposed potential and that of each individual potential is found. The results show that the transmission coefficient obtained from the 2 × 2 transfer matrix is of a lower bound on the exact transmission coefficient.
The exact analytical solutions of the Schrödinger equation for the generalized symmetrical Woods-Saxon potential are examined for the scattering, bound, and quasi-bound states in one dimension. The reflection and transmission coefficients are analytically obtained. Then, the correlations between the potential parameters and the reflection-transmission coefficients are investigated, and a transmission resonance condition is derived. Occurrence of the transmission resonance has been shown when incident energy of the particle is equal to one of the resonance energies of the quasi-bound states.
We show that the δ-function potential can be exploited along with perturbation theory to yield the result of certain infinite series. The idea is that any exactly soluble potential, if coupled with a δ function potential, remains exactly soluble. We use the strength of the δ function as an expansion parameter and express the second-order energy shift as an infinite sum in perturbation theory. The analytical solution is used to determine the second-order energy shift and hence the sum of an infinite series. By an appropriate choice of the unperturbed system, we can show the importance of the continuum in the energy shift of bound states.