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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Boundary conditions in the Aharonov-Bohm scattering of Dirac particles and the effect of Coulomb interaction

Francisco Antônio Bezerra Coutinho, J. Fernando Perez

Published Jul 15, 1993• 17 citations
Publisher

Abstract

We consider the question of the physically correct choice of the boundary conditions, and therefore of the dynamics, for the scattering of spin-\textonehalf{} particles by a thread of magnetic flux. It is shown that even when the problem is approached by first considering a source of finite radius $R$, the resulting dynamics is a consequence of how much penetration of the wave function inside the tube one chooses to allow. (The indeterminacy in the boundary condition occurs for one value of the total angular momentum for a given value of the flux.) If a Coulomb interaction of the particle with the sources is introduced in the problem, the above-mentioned indeterminacy is enlarged. We show that for a given value of the flux there are now at least two values of the angular momentum $n$ for which the indeterminacy occurs.

DOI
10.1103/physrevd.48.932

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We consider the question of the physically correct choice of the boundary conditions, and therefore of the dynamics, for the scattering of spin-\textonehalf{} particles by a thread of magnetic flux. It is shown that even when the problem is approached by first considering a source of finite radius RRR, the resulting dynamics is a consequence of how much penetration of the wave function inside the tube one chooses to allow. (The indeterminacy in the boundary condition occurs for one value of the total angular momentum for a given value of the flux.) If a Coulomb interaction of the particle with the sources is introduced in the problem, the above-mentioned indeterminacy is enlarged. We show that for a given value of the flux there are now at least two values of the angular momentum nnn for which the indeterminacy occurs.
The ((2+1)-dimensional) Aharonov–Bohm effect is analyzed for a spin-1/2 particle in the case that a 1/rpotential is present. Scalar and vector couplings are each considered. It is found that the approach in which the flux tube is given a finite radius that is taken to zero only after a matching of boundary conditions does not give physically meaningful results. Specifically, the operations of taking the limit of zero flux tube radius and the Galilean limit do not commute. Thus there appears to be no satisfactory solution of the relativistic Aharonov–Bohm–Coulomb problem using the finite radius flux tube method.
We show that the Hamiltonian H and the helicity operator Λ of a Dirac particle moving in two dimensions in the presence of an infinitely thin magnetic flux tube each admit a four-parameter family of self-adjoint extensions. Each extension is in one-to-one correspondence with the boundary conditions (BCs) to be satisfied by the eigenfunctions at the origin. Although the actions of these two operators commute before specification of BCs, to ensure helicity conservation it is not sufficient to take the same BCs for both operators. We show that, given certain relations between the parameters of the extensions, it is possible to write down the most general domain where both operators H and Λ are self-adjoint with helicity conservation and also Aharonov-Bohm symmetry (ϕ→ϕ + 1) preserved, where ϕ is the magnetic flux in natural units. The continuity of the dynamics is also obtained. Our results imply that neither helicity conservation nor Aharonov-Bohm symmetry by itself solves the problem of choosing the `physical' BCs for this system.
We consider the Dirac equation in cylindrically symmetric magnetic fields and find its normal modes as eigenfunctions of a complete set of commuting operators. This set consists of the Dirac operator itself, the zzz-components of the linear and the total angular momenta, and of one of the possible spin polarization operators. The spin structure of the solution is completely fixed independently of the radial distribution of the magnetic field which influences only the radial modes. We solve explicitly the radial equations for the uniform magnetic field inside a solenoid of a finite radius and consider in detail the scattering of scalar and Dirac particles in this field. For particles with low energy the scattering cross section coincides with the Aharonov-Bohm scattering cross section. We work out the first order corrections to this result caused by the fact that the solenoid radius is finite. At high energies we obtain the classical result for the scattering cross section.
We consider the Dirac equation in cylindrically symmetric magnetic fields and find its normal modes as eigenfunctions of a complete set of commuting operators. This set consists of the Dirac operator itself, the z-components of the linear and the total angular momenta, and of one of the possible spin polarization operators. The spin structure of the solution is completely fixed independently of the radial distribution of the magnetic field which influences only the radial modes. We solve explicitly the radial equations for the uniform magnetic field inside a solenoid of finite radius and consider in detail the scattering of scalar and Dirac particles in this field. For particles with low energy the scattering cross section coincides with the Aharonov - Bohm scattering cross section. We work out the first-order corrections to this result caused by the fact that the solenoid radius is finite. At high energies we obtain the classical result for the scattering cross section.
The basic AB problem is to determine how an unshielded tube of magnetic flux $\ensuremath{\Phi}$ affects arbitrarily long-wavelength charged particles impinging on it. For spin 1 at almost all $\ensuremath{\Phi}$ the particles do not penetrate the tube, so the interaction essentially is periodic in $\ensuremath{\Phi}$ (AB effect). Below-threshold bound states move freely only along the tube axis, and consequent induced vacuum currents supplement rather than screen $\ensuremath{\Phi}$. For a pure magnetic interaction the tube must be broader than the particle Compton wavelength, i.e., only the nonrelativistic spin-111 AB problem exists.
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.