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arXiv:hep-th/0201068 · Journal of Mathematical Physics

Solutions of relativistic wave equations in superpositions of Aharonov–Bohm, magnetic, and electric fields

Vladislav G. Bagrov, Dmitri M Gitman, V. B. Tlyachev

Published May 1, 2001• 20 citations
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Abstract

We present new exact solutions (in 3+1 and 2+1 dimensions) of relativistic wave equations (Klein–Gordon and Dirac) in external electromagnetic fields of special form. These fields are combinations of Aharonov–Bohm solenoid field and some additional electric and magnetic fields. In particular, as such additional fields, we consider longitudinal electric and magnetic fields, some crossed fields, and some special nonuniform fields. The solutions obtained can be useful to study the Aharonov–Bohm effect in the corresponding electromagnetic fields.

DOI
10.1063/1.1353182

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We present new exact solutions (in 3+1 and 2+1 dimensions) of relativistic wave equations (Klein–Gordon and Dirac) in external electromagnetic fields of special form. These fields are combinations of Aharonov–Bohm solenoid field and some additional electric and magnetic fields. In particular, as such additional fields, we consider longitudinal electric and magnetic fields, some crossed fields, and some special nonuniform fields. The solutions obtained can be useful to study the Aharonov–Bohm effect in the corresponding electromagnetic fields.
The structure of additional electromagnetic fields to the Aharonov–Bohm field, for which the Schrödinger, Klein-Gordon, and Dirac equations can be solved exactly are described and the corresponding exact solutions are found. It is demonstrated that aside from the known cases (a constant and uniform magnetic field that is parallel to the Aharonov–Bohm solenoid, a static spherically symmetrical electric field, and the field of a magnetic monopole), there are broad classes of additional fields. Among these new additional fields we have physically interesting electric fields acting during a finite time or localized in a restricted region of space. There are additional time-dependent uniform and isotropic electric fields that allow exact solutions of the Schrödinger equation. In the relativistic case there are additional electric fields propagating along the Aharonov–Bohm solenoid with arbitrary electric pulse shape.
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 analytic solutions are found to the Dirac equation in 2+12+12+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.
It is shown that the vector potential of the Aharonov-Bohm effect is effectively the same as vector potentials of the quantum phases of electric and magnetic dipoles such as the Aharonov-Casher effect, when the Aharonov-Bohm solenoid is a straight cylinder with symmetric cross sections. With a circular cross section, the Aharonov-Bohm vector potential is effectively the same as the Aharonov-Casher vector potential, while with a rectangular cross section, it is effectively the same as the vector potential of the Casella phase. Therefore, we may say that the quantum phases of the dipoles belong to a subclass of the Aharonov-Bohm phase, and they must share the topological origin of the Aharonov-Bohm phase, even though their topological freedom is limited to the particle trajectories. However, the dipole phases have the extra relativistic effect that a magnetic (electric) dipole moving in an electric (magnetic) field feels an additional magnetic (electric) field, in its rest frame. This effect limits further the topological freedom of the dipole phases.
The two-dimensional solution of the spinless Klein–Gordon (KG) equation for scalar–vector harmonic oscillator potentials with and without the presence of constant perpendicular magnetic and Aharonov–Bohm (AB) flux fields is studied within the asymptotic function analysis and Nikiforov–Uvarov (NU) method. The exact energy eigenvalues and normalized wave functions are analytically obtained in terms of potential parameters, magnetic field strength, AB flux field and magnetic quantum number. The results obtained by using different Larmor frequencies are compared with the results in the absence of both magnetic field (ωL = 0) and AB flux field (ξ = 0) case. Effects of external fields on the non-relativistic energy eigenvalues and wave functions solutions are also precisely presented.