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TURKISH JOURNAL OF PHYSICS

Bound State Aharonov-Bohm Effect and Anyons

ABDULLAH VERÇİN, NUSRET GÜÇLÜ

Published Jan 1, 1997• 0 citations
Publisher

Abstract

The effects of a standing solenoid in the energy and total angular momentum eigenstates of $N$ anyon moving in a static and uniform magnetic field are investigated. It has been shown that, without changing the statistics of the eigenfunctions, a kind of statistical transmutation in the spectra is possible by adjusting the flux of the solenoid. Keywords: Aharonov-Bohm Effect, Anyons, Transmutation of the Statistics.

DOI
10.55730/1300-0101.2418

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The effects of a standing solenoid in the energy and total angular momentum eigenstates of NNN anyon moving in a static and uniform magnetic field are investigated. It has been shown that, without changing the statistics of the eigenfunctions, a kind of statistical transmutation in the spectra is possible by adjusting the flux of the solenoid. Keywords: Aharonov-Bohm Effect, Anyons, Transmutation of the Statistics.
The role of the Aharonov-Bohm effect is analyzed for the bound states of a charged particle in the superposition of two fields: A uniform magnetic field plus the vector potential of a solenoid. Exact solutions are given for a solenoid of zero size and the corrections due to finite size are discussed. It is shown that a series of new "betatron resonances" appear in the spectrum, dependent on the flux quantum number $f=\frac{e\ensuremath{\Phi}}{\mathrm{hc}}$. The fundamental betatron resonance remains narrow and unshifted for solenoid radii large enough to broaden and shift the cyclotron resonance.
In quantum mechanics, the magnetic field has a physical effect beyond its local presence. For axially symmetric vector potentials, A=A(s,z)ϕ̂, it is the magnetic flux that plays a central role in both interference experiments (as in the Aharonov-Bohm effect) and in bound state spectra for a particle constrained to move in a circle. The flux dependence is evident in various configurations, and we have chosen two applications to highlight the non-local nature of the quantum mechanical magnetic field dependence: A finite solenoid and a Dirac string with a non-zero radius. The finite solenoid is interesting because it represents the actual experimental apparatus for measuring the Aharonov-Bohm effect. The Dirac string provides an electric charge quantization mechanism that depends on where the quantization argument is applied, yielding different fundamental charge units for different locations.
The Aharonov–Bohm effect can be described as a phase difference in interfering charged particles that travel through two distinct pathways oppositely surrounding a perpendicularly-positioned solenoid. The magnetic field emanates from the solenoid but does not intersect the pathways. On the other hand, the Kuramoto model can be used to identify the synchronization conditions that lead to a particular phase difference by treating the phases as coupled oscillators. Starting with the overall wave function expression for the electron in an Aharonov–Bohm potential, we derive a version of the Kuramoto model describing the phase dynamics of the bound state of the quantum mechanical system. We show that the resulting synchronization condition of the model coincides with the allowable values of the flux parameter for our case to achieve an Aharonov–Bohm effect.
Aharonov-Bohm solenoid changes the energy spectrum of charge particles in pure magnetic field. In particular, the degeneracy with respect to azimuthal quantum number l is partially lifted. In turn, this complicates the radiation spectrum of a charged particle in magnetic field in the presence of the solenoid (Aharonov-Bohm effect). In particular, the degeneracy of the radiation intensity with respect to the azimuthal quantum number is lifted completely. In the present work we study l-dependence (induced by Aharonov-Bohm solenoid) of synchrotron radiation intensity in semiclassical approximation.
In the context of the Aharonov-Bohm effect, it is well-known that the local action of the vector potential or the non-local action of the magnetic field are properties attributable to quantum physics, which can explain the observed Aharonov-Bohm phase. Here we point out that, surprisingly, these properties are also attributable to classical physics and that this result follows from examining the electromagnetic angular momentum of a static electric charge outside an infinitely-long solenoid of finite radius carrying a uniform magnetic flux. This angular momentum may be expressed in three equivalent forms. The first form shows that there is an electromagnetic angular momentum even in regions for which the magnetic field vanishes but the potential does not, which stresses the physical nature of the vector potential in classical physics. The second form shows that there is an electromagnetic angular momentum produced by the non-local action of the magnetic field. The third form shows that the Aharonov-Bohm phase is proportional to the magnitude of the electromagnetic angular momentum, which sheds light on the physical meaning of this phase.
In two-dimensional space, the topological coupling arising between charged particles with attached magnetic fluxes, via the Aharonov–Bohm effect, causes distinguishable particles to effectively acquire quantum statistics: a nontrivial quantum phase is generated when a particle of one species encircles one of a different species. We discuss a number of exact and numerical results concerning distinguishable particles with fractional statistics. Just like for anyons, the spectrum of such particles confined to the lowest Landau level of a strong magnetic field can be found exactly. Then a system of particles in the presence of static magnetic fluxes (equivalent to particles with infinite mass) is considered, and the low-lying states of one particle in the presence of two fluxes and two particles in the presence of one flux are analyzed.