Physical Review A
(Atomic, Molecular, and Optical Physics)
May 1994, Volume 49, No. 5, pp. 3310-3321

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One-dimensional scattering: Recurrence relations and differential equations for transmission and reflection amplitudes

M. G. Rozman and P. Reineker
Abteilung für Theoretische Physik, Universität Ulm, D-89069 Ulm, Germany
R. Tehver
Institute of Theoretical Physics, Tartu University, EE2400 Tartu, Estonia
Received 3 November 1993

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.

©1994 The American Physical Society

PACS: 03.65.Nk, 03.65.Ca, 02.70.-c, 73.40.Gk

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© 2001 Michael Rozman (
Last modified: May 16, 2001