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American Journal of Physics

Bound states and scattering from a one-dimensional hydrogen atom at a boundary

I. Richard Lapidus

Published Dec 1, 1983• 5 citations
Publisher

Abstract

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.

DOI
10.1119/1.13330

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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.
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.
We consider bound and scattering states of the one-dimensional dimer formed by two coupled non-identical atoms when one of them also interacts with the zero-range potential located at the origin. By calculating the dimer localized and scattering wave functions, we identify properties of the system without the two-body bound-state breakdown. In general, we predict an enhancement of the dimer reflection compared to a single atom, except for a narrow region on the attractive side of the external potential.
Abstract In this work, we investigate the bound-state problem in a one-dimensional spin-1 Dirac Hamiltonian with a flat band. It is found that the flat band has significant effects on the bound states. For example, for Dirac delta potential gδ(x), there exists one bound state for both the positive and negative potential strength g. Furthermore, when the potential is weak, the bound-state energy is proportional to the potential strength g. For square well potential, the flat band results in the existence of infinite bound states for arbitrarily weak potential. In addition, when the bound-state energy is very near the flat band, the energy displays a hydrogen atom-like spectrum, i.e. the bound-state energies are inversely proportional to the square of the natural number n (e.g., E n ∝ 1/n 2, n = 1, 2, 3, …). Most of the above nontrivial behaviors can be attributed to the infinitely large density of states of the flat band and its ensuing 1/z singularity of the Green function. The combination of a short-ranged potential and flat band provides a new possibility to get an infinite number of bound states and a hydrogen atom-like energy spectrum. In addition, our findings provide some useful insights and further our understanding of the many-body physics of the flat band.
There exists a simple relationship between a quantum-mechanical bound-state wave function and that of nearby scattering states, when the scattering energy is extrapolated to that of the bound state. This relationship is demonstrated numerically for the case of a spherical well potential and analytically for this and other soluble potentials. Provided that the potential is of a finite range and that the binding is weak, the theorem gives a useful approximation for the short-distance behavior of the scattering wave functions. The connection between bound and scattering-state perturbation theory is established in this limit.