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Journal of Atomic and Molecular Physics

Effect of a magnetic flux line on the quantum level density of a harmonic oscillator

Xiao Ma

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

Using the periodic orbit theory,we computed the quantum level density of a particle in the two-dimensional harmonic oscillator potential with and without the magnetic flux line for different cases.Especially discuss the influence of the magnetic flux line on the quantum level density.The results show when the frequency ratio of the two-dimensional harmonic potential is a rational number,the quantum level density is discrete.Each peak in the level density corresponds to one energy.However,when the frequency ratio is an irrational number,the level density is oscillating,when the magnetic flux is added,the amplitude of the oscillation decreased.This can be considered as a consequence of Aharonov-Bohm effect.

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Using the periodic orbit theory,we computed the quantum level density of a particle in the two-dimensional harmonic oscillator potential with and without the magnetic flux line for different cases.Especially discuss the influence of the magnetic flux line on the quantum level density.The results show when the frequency ratio of the two-dimensional harmonic potential is a rational number,the quantum level density is discrete.Each peak in the level density corresponds to one energy.However,when the frequency ratio is an irrational number,the level density is oscillating,when the magnetic flux is added,the amplitude of the oscillation decreased.This can be considered as a consequence of Aharonov-Bohm effect.
Using Gutzwiller's periodic orbit theory, we study the quantum level density of a spherical billiard in the presence of a magnetic flux line added at its center, especially discuss the influence of the magnetic flux strength on the quantum level density. The Fourier transformed quantum level density of this system has allowed direct comparison between peaks in the level density and the length of the periodic orbits. For particular magnetic flux strength, the amplitude of the peaks in the level density decreased and some of the peaks disappeared. This result suggests that Aharonov-Bohm effect manifests itself through the cancellation of periodic orbits. This phenomenon will provide a new experimental testing ground for exploring Aharonov-Bohm effect.
We study the effects of the perpendicular magnetic and Aharonov—Bohm (AB) flux fields on the energy levels of a two-dimensional (2D) Klein—Gordon (KG) particle subjected to an equal scalar and vector pseudo-harmonic oscillator (PHO). We calculate the exact energy eigenvalues and normalized wave functions in terms of chemical potential parameter, magnetic field strength, AB flux field, and magnetic quantum number by means of the Nikiforov—Uvarov (NU) method. The non-relativistic limit, PHO, and harmonic oscillator solutions in the existence and absence of external fields are also obtained.
The quantum density of states of the Henon-Heiles potential displays a pronounced beating pattern. This has been explained by the interference of three isolated classical periodic orbits with nearby actions and periods. A singular magnetic flux line, passing through the origin, drastically alters the beats even though the classical Lagrangian equations of motion remain unchanged. Some of the changes can be easily understood in terms of the Aharonov-Bohm effect. However, we find that the standard periodic orbit theory does not reproduce the diffraction-like quantum effects on those classical orbits which intersect the singular flux line, and argue that corrections of relative order variant Planck's over 2pi are necessary to describe these effects. We also discuss the changes in the distribution of nearest-neighbor spacings in the eigenvalue spectrum, brought about by the flux line. (c) 1995 American Institute of Physics.
We discuss the application of Gutzwiller's semiclassical theory to a circular billiard with a singular magnetic flux line added at its center. The Aharonov-Bohm effect manifests itself through the cancellation of periodic orbits for particular flux strengths. Diffraction phenomena affect the gross-shell structure of the level density and require corrections of higher order in \ensuremath{\Elzxh}. The full quantization of the level spectrum, however, is much less affected. \textcopyright{} 1996 The American Physical Society.
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 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.