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Journal of Physics A Mathematical and General

Relativistic two-particle one-dimensional scattering problem for superposition of delta-potentials

V. N. Kapshaǐ, Tatjana Alferova

Published Jan 1, 1999• 13 citations
Publisher

Abstract

The covariant single-time equations of the quantum field theory are formulated in the relativistic configurational representation. The explicit formulae for the Green functions corresponding to the scattering states are calculated in this representation. Using the derived nonhomogeneous equations the scattering problem is solved exactly for certain potentials (combinations of zero-range potentials). The equations and their solutions are studied in the non-relativistic limit. The conditions of total reflection, available for such potentials, are investigated.

DOI
10.1088/0305-4470/32/28/311

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The covariant single-time equations of the quantum field theory are formulated in the relativistic configurational representation. The explicit formulae for the Green functions corresponding to the scattering states are calculated in this representation. Using the derived nonhomogeneous equations the scattering problem is solved exactly for certain potentials (combinations of zero-range potentials). The equations and their solutions are studied in the non-relativistic limit. The conditions of total reflection, available for such potentials, are investigated.
In this work, exact solutions are obtained for two-dimensional partial integral equations of the Logunov-Tavkhelidze type for a system of two particles of equal mass in the case of a «delta-circle» potential and a superposition of two such potentials, defined in the relativistic configuration representation. It is shown that for a single «delta-circle» potential, depending on its parameters, the system can have either one bound state or none at all. For a superposition of two «delta-circle» potentials, cases with one or two bound states, or their absence, are possible. Analysis of the wave functions revealed differences in their properties across different representations: in the relativistic configuration representation, the number of zeros of the partial wave function equals the state number of the particle system, whereas in the momentum representation, the number of zeros exceeds that in the configuration representation. The nonrelativistic limit of the obtained solutions is found, which agrees with the solutions of the two-dimensional Schrödinger equation for similar potentials in the momentum and coordinate representations.
The problem of determining a scattering potential from scattered field data is addressed within the two integral equation formulation of scattering theory first proposed by Goldberger (1964). The lowest-order approximation within this formulation casts the inverse problem entirely in terms of the Heitler equation which relates the on-shell T matrix directly to the scattering potential. This lowest-order approximation is applied to the case of non-relativistic quantum mechanical inverse scattering from one-dimensional local potentials and is found to yield an exact solution for the case of a one-dimensional delta function potential.
This paper discusses the formulation of a quantum mechanical equivalent of the relative time classical theory proposed in Part I. The relativistic wavefunction is derived and a covariant addition theorem is put forward which allows a covariant scattering theory to be established. The free particle eigenfunctions that are given are found not to be plane waves. A covariant partial wave analysis is also given. A means is described of converting wavefunctions that yield probability densities in 4-space to ones that yield the 3-space equivalents. Bound states are considered and covariant analogues of the Coulomb potential, harmonic oscillator potential, inverse cube law of force, square well potential, and two-body fermion interactions are discussed.
The scattering of relativistic spin-one-half particles in an Aharonov-Bohm potential is considered. It is shown that earlier approaches to this problem have neglected a crucial delta function contribution to the potential. By formulating the problem with a source of finite radius which is then allowed to go to zero, it is established that this is the delta function alone that causes solutions that are singular at the origin to become relevant. The changes in the amplitude which arise from the inclusion of spin are seen to modify the cross section for the case of polarized beams. Finally, the calculated Aharonov-Bohm amplitude is shown to describe the scattering of particles with arbitrary spin in the c=\ensuremath{\infty} limit.