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Journal of Jiamusi University

Re-discussion on the Bound Energy and Scattering Amplitude Pole by the Schrdinger and the Heisenberg Representation

Wang Zhi-liang

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

One-dimensional δ-potential well bound state and one-dimensional δ-scattering potential were discussed by Schrdinger and Heisenberg representation in the wave function in the time-dependent state.It is proved more generally that one-dimensional δ-potential well restraint level is the δ-barrier scattering amplitude pole.

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The bound-state energy of a one-dimensional hydrogen atom with a δ-function potential is modified by the presence of a nearby boundary. The distortion of the wave function raises the energy of the atom. If the atom is sufficiently close to a high barrier the system does not have a bound state. Scattering from a δ-function potential is also modified by the presence of the boundary. This simple example, which can be solved exactly, may help to provide insight into more realistic interactions between single atoms and surfaces.
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The time delay or the time advance in the scattering of simple one-dimensional systems can be evaluated in a straightforward manner for certain potential models. It is found that when the interacting potential is attractive and has a strength such that it nearly supports an additional bound state, the time delay at small scattering energy is very large. On the other hand, if the potential supports a bound state with nearly zero binding energy, the time advance near threshold is anomalously large. The behavior of a wave packet scattering from the double delta-function potential is also investigated.