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American Journal of Physics

Interaction of a charge and an electric dipole in one dimension

I. Richard Lapidus

Published Jan 1, 1980• 3 citations
Publisher

Abstract

The quantum-mechanical interaction of an electric charge and an electric dipole in one dimension is investigated using the delta-function potential to represent the interaction between charges. It is found that there is a single bound state. The problem of scattering of a charge from an electric dipole is solved exactly and in Born approximation. Because the potential is not symmetric, a phase shift analysis does not yield the correct cross section.

DOI
10.1119/1.12240

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The quantum-mechanical interaction of an electric charge and an electric dipole in one dimension is investigated using the delta-function potential to represent the interaction between charges. It is found that there is a single bound state. The problem of scattering of a charge from an electric dipole is solved exactly and in Born approximation. Because the potential is not symmetric, a phase shift analysis does not yield the correct cross section.
A phase-shift analysis of quantum mechanical scattering problems in one dimension is developed. Many of the techniques and results of three dimensions remain even in one dimension, while the discussion is vastly simplified, and thus accessible to the student much earlier, due to the absence of special functions (in particular, Bessel and Legendre functions) with their complicated asymptotic formulas and orthogonality relations. A “partial wave” analysis leads to the familiar relation fl∼eiδl sinδl between the scattering amplitude in the lth partial wave and the phase shift δl. The optical theorem is evident as an immediate consequence of the conservation of probability. The method is applied to an analysis of the well-known square-well scattering resonances.
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.
In this paper it is shown that several cases of one-dimensional N-body problems are exactly soluble. The first case describes the motion of three one-dimensional particles of arbitrary mass which interact with one another via infinite-strength, repulsive delta-function potentials. It is found in this case that the stationary-state solution of the scattering of the three particles is analogous to an electro-magnetic diffraction problem which has already been solved. The solution to this analogous electro-magnetic problem is interpreted in terms of particles. Next it is shown that the problem of three particles of equal mass interacting with each other via finite- but equal-strength delta-function potentials is exactly soluble. This example exhibits rearrangement and bound-state effects, but no inelastic processes occur. Finally it is shown that the problem of N particles of equal mass all interacting with one another via finite- but equal-strength delta functions is exactly soluble. Again no inelastic processes occur, but various types of rearrangements and an N-particle bound state do occur. These rearrangements and the N-particle bound state are illustrated by means of a series of sample calculations.
The time-dependent Schrodinger equation for a particle in one dimension in the field of two colliding delta potentials can be solved exactly. Assuming an initial state which is bound about the left delta potential for large negative times, the solution has been used to calculate the probabilities of finding a state bound about one of the potentials long after the collision. These probabilities are oscillating functions of the reciprocal velocity. The ionisation probability goes to zero for high collision velocities and reaches 1 / 2 for small velocities.