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Study of One Dimensional Scattering by A Two-Step Potential

Francis Stephen Uralil

Published Jan 1, 1972• 0 citations
OpenAlex

Abstract

The purpose of this investigation is to contribute to the theory of one-dimensional scattering through a thorough study of the two-step potential. For this purpose we have studied the effect of changing a simple square well potential near its edge by adding a second step. The problem is treated using both perturbational and non-perturbational methods. The order of procedure is: (1) a general introduction, (2) multiple reflections of a wave packet by an attractive potential well, (3) discussion of the- two-step potential using perturbation theory, (4) calculation of depth of potential well for zero energy resonance, (5) conclusion.

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arXiv:2504.09916•Journal of High Energy Physics•Oct 21, 2025•2 citations
The purpose of this investigation is to contribute to the theory of one-dimensional scattering through a thorough study of the two-step potential. For this purpose we have studied the effect of changing a simple square well potential near its edge by adding a second step. The problem is treated using both perturbational and non-perturbational methods. The order of procedure is: (1) a general introduction, (2) multiple reflections of a wave packet by an attractive potential well, (3) discussion of the- two-step potential using perturbation theory, (4) calculation of depth of potential well for zero energy resonance, (5) conclusion.
In this paper we extend our previous analysis of the scattering of wave packets in one dimension to the case of the square-well potential. The analytic properties of the general scattering solution are emphasized thereby making the analysis useful as introductory material for a more sophisticated S-matrix treatment. The square-well model is particularly interesting because of its application to the deuteron problem. Resonance scattering, barrier penetration, time delay, and line shape are discussed at the level of the first-year graduate student.
A bstract One-dimensional scattering of a wave packet of a relativistic fermion under a temporally oscillating electric field superimposed on a potential step is discussed by using the Furry-picture perturbation theory, where the oscillating electric field is treated as a perturbation. Reflection and transmission probabilities of the wave packet, which in its single-mode limit are consistent with those in the stationary scattering off the potential step alone, are investigated up to the second order. We show that even in the absence of the so-called Klein region, a positive-frequency incoming wave can penetrate the negative- frequency region below the potential step by emitting its energy to the oscillating electric field with a finite tunneling probability.