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Theoretical and Mathematical Physics

Pauli isotonic oscillatorwith an anomalous magnetic moment in the presence of the Aharonov–Bohm effect: Laplace transform approach

Muhammad Roshanzamir, H. Goudarzi

Published Feb 1, 2016• 2 citations
Publisher

Abstract

A strong magnetic field significantly affects the intrinsic magnetic moment of fermions. In quantum electrodynamics, it was shown that the anomalous magnetic moment of an electron arises kinematically, while it results from a dynamical interaction with an external magnetic field for hadrons (proton). Taking the anomalous magnetic moment of a fermion into account, we find an exact expression for the boundstate energy and the corresponding eigenfunctions of a two-dimensional nonrelativistic spin-1/2 harmonic oscillator with a centripetal barrier (known as the isotonic oscillator) including an Aharonov–Bohm term in the presence of a strong magnetic field. We use the Laplace transform method in the calculations. We find that the singular solution contributes to the phase of the wave function at the origin and the phase depends on the spin and magnetic flux.

DOI
10.1134/s0040577916020124

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A strong magnetic field significantly affects the intrinsic magnetic moment of fermions. In quantum electrodynamics, it was shown that the anomalous magnetic moment of an electron arises kinematically, while it results from a dynamical interaction with an external magnetic field for hadrons (proton). Taking the anomalous magnetic moment of a fermion into account, we find an exact expression for the boundstate energy and the corresponding eigenfunctions of a two-dimensional nonrelativistic spin-1/2 harmonic oscillator with a centripetal barrier (known as the isotonic oscillator) including an Aharonov–Bohm term in the presence of a strong magnetic field. We use the Laplace transform method in the calculations. We find that the singular solution contributes to the phase of the wave function at the origin and the phase depends on the spin and magnetic flux.
Invariant operator method for discrete or continuous spectrum eigenvalue and unitary transformation approach are employed to study the two-dimensional time-dependent Pauli equation in presence of the Aharonov–Bohm effect (AB) and external scalar potential. For the spin particles the problem with the magnetic field is that it introduces a singularity into wave equation at the origin. A physical motivation is to replace the zero radius flux tube by one of radius R, with the additional condition that the magnetic field be confined to the surface of the tube, and then taking the limit R → 0 at the end of the computations. We point that the invariant operator must contain the step function θ(r – R). Consequently, the problem becomes more complicated. In order to avoid this difficulty, we replace the radius R by ρ(t)R, where ρ(t) is a positive time-dependent function. Then at the end of calculations we take the limit R → 0. The qualitative properties for the invariant operator spectrum are described separately for the different values of the parameter C appearing in the nonlinear auxiliary equation satisfied by ρ(t), i.e., C > 0, C = 0, and C < 0. Following the C's values the spectrum of quantum states is discrete (C > 0) or continuous (C ≤ 0).
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.
In the nonrelativistic or relativistic studies of the well-known effect of an infinitely long and infinitely thin solenoid containing magnetic flux Φ, the so-called Aharonov-Bohm (AB) field, in terms of the Dirac's fundamental unit of the magnetic flux Φ 0 , the magnetic flux Φ is usually expressed as a flux quantum number, which can be decomposed into two parts: the integer and the mantissa part. It has been commonly accepted in the literature that the integer part is unimportant and only the mantissa determines all the physical effects. In this paper, by investigating the AB effect on the the sojourn time of a two-dimensional (2D) inverted harmonic oscillator under a perpendicular uniform magnetic field, we show that the sojourn time depends upon the magnitude, rather than solely the mantissa of the magnetic flux quantum number. This then provides an example that can manifest the quantum effect of the integer part of the flux quantum number, the reason behind is explained via gauge transformation.
The scattering of a nonrelativistic neutral massive fermion having the anomalous magnetic moment (AMM) in an electric field of a uniformly charged long conducting thread aligned perpendicularly to the fermion motion is considered to study the so-called Aharonov–Casher (AC) effect by taking into account the particle spin. For this solution, the nonrelativistic Dirac–Pauli equation for a neutral massive fermion with AMM in (3+1) dimensions is found, which takes into account explicitly the particle spin and interaction between AMM of moving fermion and the electric field. Expressions for the scattering amplitude and the cross-section are obtained for spin-polarized massive neutral fermion scattered off the above conducting thread. We conclude that the scattering amplitude and cross-section of spin-polarized massive neutral fermions are influenced by the interaction of AMM of moving neutral fermions with the electric field as well as by the polarization of fermion beam in the initial state.
The Aharonov-Bohm scattering of relativistic spin-1∕21∕21∕2 particles with an anomalous magnetic moment are studied. The scattering cross sections for unpolarized and polarized particles are obtained by solving the Dirac-Pauli equation. It is somewhat unexpected that the results are in general the same as those for particles without an anomalous magnetic moment. However, when the incident energy takes some special values, the cross section for polarized particles is dramatically changed. In these cases the helicity of scattered particles is not conserved. In particular, the helicity of particles scattered in the backward direction is all reversed. In the nonrelativistic limit, a very simple relation between the polarized directions of the incident and scattered particles is found, for both general and special incident energies. For particles without an anomalous magnetic moment this relation can be drawn from previous results but it appears to be unnoticed.