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Journal of Contemporary Physics (Armenian Academy of Sciences)

Amplitudes of two-channel scattering by a two-dimensional potential barrier with a constant height

Д. М. Седракян, Eduard M Kazaryan, Л. Р. Седракян

Published Apr 29, 2010• 5 citations
Publisher

Abstract

The immersing method is used to solve the two-channel scattering problem in the case of concrete potential. In particular, we consider the particle scattering by a two-dimensional barrier which is constant in the scattering direction and is arbitrary in the transverse direction. For this case the scattering amplitudes t 1, t 2, r 1, and r 2 are determined. Expressions for the transmission and reflection amplitudes are obtained for the case of δ-potential. The behavior of the scattering amplitudes in the limit k 2 → 0 is studied. It is also shown that the ratio of products of transmission and reflection amplitudes for two channels does not depend on the coordinate of the middle of coordinate.

DOI
10.3103/s1068337210030035

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The immersing method is used to solve the two-channel scattering problem in the case of concrete potential. In particular, we consider the particle scattering by a two-dimensional barrier which is constant in the scattering direction and is arbitrary in the transverse direction. For this case the scattering amplitudes t 1, t 2, r 1, and r 2 are determined. Expressions for the transmission and reflection amplitudes are obtained for the case of δ-potential. The behavior of the scattering amplitudes in the limit k 2 → 0 is studied. It is also shown that the ratio of products of transmission and reflection amplitudes for two channels does not depend on the coordinate of the middle of coordinate.
The immersing method is applied to solve the N-channel scattering problem for concrete potential. In particular, we consider the particle scattering on a two-dimensional potential barrier, which is constant in the scattering direction and arbitrary in the cross-section direction. For this case the scattering amplitudes t m and r m (1,2, m= ...,N) are determined. A transition from the obtained formulas to the case of ™-potential is performed. For this case transmission amplitudes t m and reflection amplitudes r m are obtained. It is also shown that the product of transmission and reflection amplitudes along the channel m does not depend on the scattering channel.
The immersing method is applied to solve the N-channel scattering problem. In particular, we consider the particle scattering on a multidimensional potential barrier, which is constant in the scattering direction and arbitrary in the lateral direction. For this case the scattering amplitudes t m and r m (m = 1, 2, …, N) are determined. Transition from the obtained formulas to the case of thin potential is performed. For this case analytical expressions of transmission amplitudes t m and reflection amplitudes r m are obtained. We show that the product of transmission and reflection amplitudes in the channel m does not depend on the scattering channel. It is assumed that the scattering particle falls on the potential with the longitudinal wave vector k l corresponding to the channel l.
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.
The quasi-one-dimensional scattering of a quantum particle on the two-dimensional δ-potential is considered. Analytical expressions for multi-channel transmission t n and reflection r n amplitudes are given. The problem is solved for the case of finite number of channels that is equal to N when the particle is incident on the potential while moving in l channel. The case of three-channel scattering was examined in details. As was shown within the frameworks of this problem, the “overpopulation” of particles in the second and third channels occurs when k 2 → 0 and k 3 → 0. The location points of δ-potential that provide the complete “overpopulation” of channels were also obtained.
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.