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arXiv:quant-ph/9710049 · American Journal of Physics

Bound and unbound wave functions at short distances

Göran Fäldt A, Colin Wilkin C

Published Oct 1, 1998• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1119/1.18987

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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.
We study the structure of bound states appearing in systems governed by the Coulomb and short-range interactions. We analyze the binding energies and wave functions of the bound states generated by the Coulomb plus short-range potential. We demonstrate that Coulomb-induced shifts of the binding energy are closely correlated with the spatial distribution of the wave function. Furthermore, we show that the asymptotic behavior of wave functions of weakly bound states is qualitatively altered by Coulomb repulsion, leading to a modification of the near-threshold mass scaling that is otherwise universal for short-range interactions.
The problem of determining the Schr\"odinger wave function of a nonrelativistic particle that is either scattered by a potential of a finite range or that is bound to it is reformulated in a novel way. It is shown that in either case the wave function must satisfy a certain boundary condition on the surface that delimits the effective range of the potential. For scattering states the boundary condition is analogous to the mathematical formulation of the Ewald-Oseen extinction theorem of classical electromagnetic theory. The new formulation is illustrated by determining the scattering states and the bound states for a central potential. It is also shown that a boundary condition that is used in band-structure calculations in solids is an immediate consequence of our quantum-mechanical extinction theorem for bound states.
Abstract We discuss a general description of the solutions to the 1D time-independent Schrödinger equation that does not a priori distinguish between scattering states and bound states and emphasizes and reinforces their relationship and connection to each other. This manuscript also introduces the concept of transfer matrices, which it presents as a logical extension of the traditional approach to evaluating 1D potentials. Using the transfer matrix method and a finite step approximation allows for a simple and straight-forward numerical solution of arbitrary 1D potentials. It also separates the process of solving the Schrödinger equation from selecting physically relevant solutions, which is a critical skill in quantum mechanics and is at the core of physics problems in general.
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.
Abstract We proceed in this chapter to discuss scattering theory and related aspects of the bound-state problem in quaternionic quantum mechanics. We begin with a detailed discussion of a simple one-dimensional example, the delta function potential model. From this model will emerge the result (Adler, 1988) that will be the principal theme of this and the succeeding two chapters: with the standard ray representative choice of Sec. 4.2, for short-range potentials the asymptotic scattering states and the S-matrix in quaternionic quantum mechanics are always complex ℂ(1, i).