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Journal of Physics A Mathematical and Theoretical

The {\triangledown}^{\!\!\!\!\!\!\smash{\raise2pt\hbox{$\ssty\rightharpoonup$}}} \cdot \skew2\vec{E} term does not change the Aharonov–Casher scattering cross section

M. S. Shikakhwa, E. AL-QAQ

Published Aug 12, 2010• 3 citations
Publisher

Abstract

The Aharonov–Casher (AC) Hamiltonian with the inclusion of the term, which introduces a new singular interaction at the origin, is considered. Proceeding within the framework of the model of a finite-radius filament that is shrunk to zero at the end, the effect of the inclusion of this term on the original AC cross section is investigated. While the inclusion of this term gives rise to components of the radial wavefunctions that are irregular at the origin, it is found that the original AC cross section is the same whether one includes it or not, and regardless of whether the resulting singular interaction is attractive or repulsive. The underlying physics is discussed, and the physical equivalence with the spin one-half AB case is established.

DOI
10.1088/1751-8113/43/35/354008

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The Aharonov–Casher (AC) Hamiltonian with the inclusion of the term, which introduces a new singular interaction at the origin, is considered. Proceeding within the framework of the model of a finite-radius filament that is shrunk to zero at the end, the effect of the inclusion of this term on the original AC cross section is investigated. While the inclusion of this term gives rise to components of the radial wavefunctions that are irregular at the origin, it is found that the original AC cross section is the same whether one includes it or not, and regardless of whether the resulting singular interaction is attractive or repulsive. The underlying physics is discussed, and the physical equivalence with the spin one-half AB case is established.
In the Aharonov–Casher (AC) scattering, a neutral particle interacts with an infinitesimally thin, long charge filament resulting in a phase shift. In the original AC treatment, a ∇ · E term proportional to the charge density at the filament's position is dropped from the Hamiltonian on the basis that the particle is banned from the filament, thus, the resulting Hamiltonian compares with the Aharonov–Bohm Hamiltonian of a scalar particle. Here, we consider AC scattering with this term included. Starting from the three-dimensional nonrelativistic Aharonov–Casher (AC) Schrödinger equation with the ∇ · E term included, we find the wave functions — in particular their singular component — the phase shifts and thus compute the scattering amplitudes and cross-sections. We show that singular solutions in the AC case appear only when the delta function interaction introduced is attractive regardless of the spin orientation of the particle. We find that the inclusion of this term does not introduce a structural difference in the general form of the cross-section even for polarized particles. Its mere effect, is in shifting the parameter N (the greatest integer in α) that appears in the cross-section, in the attractive case, by one. Interesting situation appears when N = 0, thus α=δ, in the case α≻0, and N = -1, so α = 1-δ in the case α≺0: At these values of the parameter N, where αis just any fraction, the cross-section for a particle polarized in the scattering plane to scatter in a state with the same polarization, is isotropic. It also vanishes, at these values of N, for transitions between same-helicity eigenstates. For these values of the parameter N and at the special values α = ±1/2, the cross-sections for both signs of α coincide. The main differences between this model and the "mathematically equivalent" spin-1/2 AB theory are outlined.
The scattering of relativistic spin-1/2 neutral particles with a magnetic dipole moment by a long straight charged line and a magnetic flux line at the same position is studied. The scattering cross sections for unpolarized and polarized particles are obtained by solving the Dirac-Pauli equation. The results are in general the same as those for pure Aharonov-Casher scattering (by the charged line alone) as expected. However, in special cases when the incident energy, the line charge density, and the magnetic flux satisfy some relations, the cross section for polarized particles is dramatically changed. Relations between the polarization of incident particles and that of scattered ones are presented, both in the full relativistic case and the nonrelativistic limit. The characteristic difference between the general and special cases lies in the backward direction: in the general cases the incident particles are simply bounced while in the special cases their polarization is turned over simultaneously. For pure Aharonov-Casher scattering there exist cases where the helicities of all scattered particles are reversed. This seems to be remarkable but appears unnoticed previously. Two mathematical approaches are employed to deal with the singularity of the electric and magnetic field and it turns out that the physical results are essentially the same.
The problem of the proper inclusion of spin in Aharonov—Bohm scattering is considered. It is proposed that this should be accomplished by imposing the requirement that all singularities arising from the presence of spin in the associated wave equations be interpreted as limits of physically realizable flux distributions. This leads to results which confirm the usual cross section in the spinless case but imply nontrivial modifications for the scattering of a polarized spin one-half beam. By applying the technique to a calculation of the virial coefficient for a collection of flux carrying spin one-half particles, some severe obstacles to conventional views of the flux as a parameter which interpolates between bosonic and fermionic statistics are shown to occur. Although similar results for the scattering of arbitrary spin particles obtain in the Galilean limit, it is found that when spin one is considered in the context of a relativistic wave equation the singularity structure is too pathological to yield a consistent interpretation. The exact equivalence of the spin one-half Aharonov-Bohm effect to the Aharonov-Casher effect is also demonstrated and corresponding results for polarized beams are presented. Finally, it is shown that the Aharonov-Bohm effect for arbitrary spin in the Galilean limit is the exact solution in the two-particle sector of a Galilean covariant field theory.
It is shown that there is an exact equivalence between the Aharonov-Bohm effect for spin-1/2 particles and the Aharonov-Casher effect. The demonstration of this precise relationship between the two is seen furthermore to be independent of whether relativistic or nonrelativistic kinematics are used. The only remaining substantive distinction between the two effects may well be the fact that the scattering cross section for polarized beams has a considerably greater sturcture in the Aharonov-Casher case despite the mathematical equivalence of the scattering amplitudes for the two effects.
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.
The perturbation theory expansion of the Aharonov-Bohm scattering amplitude has previously been studied in the context of quantum mechanics for spin-0 and spin-1/2 particles as well as in Galilean covariant field theory. This problem is reconsidered in the framework of the model in which the flux line is considered to have a finite radius which is shrunk to zero at the end of the calculation. General agreement with earlier results is obtained but with the advantage of a treatment which unifies all the various subcases.