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American Journal of Physics

Scattering by a finite periodic potential

Donald W. L. Sprung, Hua Wu, Joan Martorell

Published Dec 1, 1993• 169 citations
Publisher

Abstract

The problem of scattering in one dimension by a potential which consists of N identical cells is solved in a transparent manner. The N-cell transmission and reflection amplitudes are expressed in terms of the single-cell amplitudes and the Bloch phase. As examples the results are applied to a row of delta-function potentials, and to a row of square wells, and it is shown that these expressions provide an immediate understanding of the results of detailed calculations.

DOI
10.1119/1.17306

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The problem of scattering in one dimension by a potential which consists of N identical cells is solved in a transparent manner. The N-cell transmission and reflection amplitudes are expressed in terms of the single-cell amplitudes and the Bloch phase. As examples the results are applied to a row of delta-function potentials, and to a row of square wells, and it is shown that these expressions provide an immediate understanding of the results of detailed calculations.
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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 problem of NNN particles interacting in one dimension by a repulsive delta-function potential is solved, when the wave function $\ensuremath{\psi}$ transforms like any irreducible representation ${R}_{\ensuremath{\psi}}$ of SN{S}_{N}SN​ for which the Young tableau consists of a finite number of either rows or columns.