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Physical Review A

Nonlocal approach to scattering in a one-dimensional problem

Ph. Grossel, J. M. Vigoureux, Fadi Baida

Published Nov 1, 1994• 27 citations
Publisher

Abstract

An alternative approach to the one-dimensional scattering problem is presented. We show that the reflection and the transmission coefficients of any number of quantum wells or barriers can directly be obtained by using two mathematical methods. The first method arises from a generalization of the so-called elementary symmetric functions used in the mathematical theory of polynomials; the second one consists of a complex generalization of Einstein's addition law for parallel velocities. These laws for the reflection and transmission amplitude coefficients permit a rapid evaluation of the transmittivity of any kind of potential barrier. Some examples are considered. Results are compared with the ones obtained with the WKB method. We also present a numerical study to test the stability of the method.

DOI
10.1103/physreva.50.3627

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An alternative approach to the one-dimensional scattering problem is presented. We show that the reflection and the transmission coefficients of any number of quantum wells or barriers can directly be obtained by using two mathematical methods. The first method arises from a generalization of the so-called elementary symmetric functions used in the mathematical theory of polynomials; the second one consists of a complex generalization of Einstein's addition law for parallel velocities. These laws for the reflection and transmission amplitude coefficients permit a rapid evaluation of the transmittivity of any kind of potential barrier. Some examples are considered. Results are compared with the ones obtained with the WKB method. We also present a numerical study to test the stability of the method.
Using the well-known matrix formulation of the reflection and transmission of electromagnetic waves by a stratified planner structure, we show that the reflection and transmission coefficients of any number of isotropic media can be written by a simple general formula. This formula uses the so-called elementary symmetric functions that are extensively used in the mathematical theory of polynomials. The approach is then applied to the quantum scattering. We show that the reflection and transmission coefficients of any number of quantum wells or barriers can be written in the similar way. Finally, one-dimensional scattering from a series of delta-function barriers (a system that is called Dirac Comb) is studied. The computed numerical illustrations compared with the earlier results based on the transfer matrix and Chebychev polynomials reveal an excellent agreement.
By calculating the reflection and transmission coefficients for a square potential barrier in terms of a series of scatterings at step potential discontinuities, a multiple scattering approach to potential problems is presented. This approach is applied to WKB potential problems by interpreting the WKB connection formulas in terms of reflection and transmission coefficients at each turning point. An algorithm for the systematic construction of the energy Green’s function is given. The significance of complex coordinate turning points is demonstrated. The equivalence of this approach and that of the Gaussian approximation of the path integral is mentioned briefly.
We show that application of the immersing and transfer-matrix methods to one-dimensional problems of particles scattering leads to the system of two linear equations for the functions F and Φ expressed by means of the transmission and reflection amplitudes. The expressions of these functions are derived. The offered method is illustrated by the finding of transmission and reflection coefficients for the potential barrier with a constant height. The developed method can be applied in solving the quasi-one-dimensional and two-dimensional problems of scattering.
A recurrence method for analytical and numerical evaluation of tunneling, transmission, and reflection amplitudes is developed. As the first step, a rule for composition of two arbitrary scatterers separated by a region of constant potential is obtained. Transmission and reflection amplitudes for this double-barrier potential are expressed in terms of transmission and reflection amplitudes for its subparts. As the length of the constant-potential region can be arbitrary and the subparts of a potential may, in turn, be arbitrary segmented potentials, one obtains formulas recurerence formulas which express the scattering amplitudes for the arbitrary segmented potential via the scattering amplitudes for the subparts into which the complete potential can be divided. The efficiency of the method is demonstrated by solving analytically the problem of scattering from locally periodic potentials. Since an arbitrary potential can be approximated by a set of infinitely narrow rectangular barriers, the recurrence formulas can be applied to any potential, giving, in the limit of zero-width segments, differential equations for transmission, and reflection amplitudes.