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arXiv:quant-ph/0406033 · Physical Review A

Relativistic Aharonov-Bohm effect in the presence of planar Coulomb potentials

V. R. Khalilov

Published Jan 19, 2005• 47 citations
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Abstract

Exact analytic solutions are found to the Dirac equation in $2+1$ dimensions for a combination of an Aharonov-Bohm potential and the Lorentz three-vector and scalar Coulomb potentials. By means of the solutions obtained the relativistic quantum Aharonov-Bohm effect is studied for the free (in the presence of a Lorentz three-vector Coulomb potential) and bound fermion states. We obtain the total scattering amplitude in a combination of the Aharonov-Bohm and Lorentz three-vector Coulomb potentials as a sum of two scattering amplitudes. This modifies the expression for the standard Aharonov-Bohm cross section due to the interference of these two amplitudes with each other. We discuss that the observable quantities can be the phases of electron wave functions or the energies of bound states.

DOI
10.1103/physreva.71.012105

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The Aharonov-Bohm scattering of massive spin-one particles is computed. Starting from the relativistic wave equation for a charged, massive spin-one particle, a reduction to two spatial dimensions is made, consistent with the standard Aharonov-Bohm (AB) geometry. It is found that the problem consists of the independent scattering of the three vector-meson spins. While the amplitude for the spin-0 mode is trivially identical to the AB amplitude, the same result is obtained for the spin-up and spin-down components, in sharp contrast to the corresponding result for the spin-1/2 AB amplitude. This can be traced to the fact that the Hamiltonian is unbounded as a consequence of the presence of a \ensuremath{\delta}-function term in the potential. In the Galilean limit it is found that the amplitude of the nonzero-spin components is identical to the spin-1/2 amplitude in that limit, in agreement with a general claim made earlier by one of the authors. The Galilean limit of the spin-one AB cross section is found to be modified and calculations for some pure-state polarizations are presented.
The authors establish the exact solution of the Dirac equation for a charged particle moving in the Coulomb field plus Aharonov-Bohm field plus Dirac monopole field by using the equation in two-dimensional complex space. On the basis of the found solution, the relativistic analogy of the Aharonov-Bohm effect for the system of a diogen atom existing in an infinite cylindrical solenoid is studied. In this case the authors obtain some surprising results. One of them is that the Aharonov-Bohm effect is absent for certain states.
Exact Green’s function related to a Dirac particle submitted to the combination of Aharonov–Bohm and Coulomb potentials in [Formula: see text]) coordinate space is analytically calculated via path integral formalism. The Pauli matrices which describe the spin dynamics are replaced by two fermionic oscillators via the Schwinger model. The energy spectrum as well as the corresponding normalized wave functions are extracted following this approach. The interesting properties of the spinors are thus deduced after symmetrization. According to the symmetric form for Green’s function, it is shown that the non-relativistic limit of the Dirac particle is undertaken with much ease.
We investigate the scattering of a charged quantum particle in a helically twisted background that induces an effective Coulomb-like interaction, in the presence of an Aharonov-Bohm (AB) flux. Starting from the nonrelativistic Schrödinger equation in the twisted metric, we derive the radial equation and show that, after including the AB potential, it can be mapped onto the same Kummer-type differential equation that governs the planar 2D2D2D Coulomb +++ AB problem, with a geometry-induced Coulomb strength and the azimuthal quantum number shifted as m→m−λm\to m-λm→m−λ. We construct the exact scattering solutions, obtain closed expressions for the partial-wave SSS matrix and phase shifts, and derive the corresponding scattering amplitude, differential cross section, and total cross section. We also show that the pole structure of the SSS matrix is consistent with the bound-state quantization previously obtained for the helically twisted Coulomb-like problem.