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arXiv:chao-dyn/9312002 · Chaos An Interdisciplinary Journal of Nonlinear Science

Effect of a magnetic flux line on the quantum beats in the Hénon–Heiles level density

M. Brack, R. K. Bhaduri, J. Law, Ch. Maier, M. V. N. Murthy

Published Mar 1, 1995• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.166080

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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.
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.
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.
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.
Motion of a nonrelativistic particle on a cone with a magnetic flux running through the cone axis (a ``flux cone'') is studied. It is expressed as the motion of a particle moving on the Euclidean plane under the action of a velocity-dependent force. The probability fluid (``quantum flow'') associated with a particular stationary state is studied close to the singularity, demonstrating nontrivial Aharonov-Bohm effects. For example, it is shown that, near the singularity, quantum flow departs from classical flow. In the context of the hydrodynamical approach to quantum mechanics, quantum potential due to the conical singularity is determined, and the way it affects quantum flow is analyzed. It is shown that the winding number of classical orbits plays a role in the description of the quantum flow. The connectivity of the configuration space is also discussed.
The authors display complex wavefunctions psi j for the jth eigenstate of a particle moving freely in a domain D (whose reflecting boundary gives classically chaotic motion) threaded by a single line of magnetic flux whose strength (in quantum units) is alpha . The wavefronts (phase contours of psi j ) show the expected dislocation singularities both away from the flux line (where their strength is +or-1) and at the flux line (where the strength is the integer closest to alpha ). psi 5 shows two dislocations away from the flux line for all alpha , and the birth of a dislocation at the flux line as alpha passes the value 1/2. psi 50 shows 43 dislocations, in reasonable agreement with a semiclassical theory based on regarding psi N as a complex Gaussian random function, which predicts N dislocations in the asymptotic limit N to infinity .
The Aharonov-Bohm effect is a physical phenomenon in which the quantum state of a charged particle acquires a phase shift that is directly proportional to the magnetic flux, ΦΦΦ, due to a (classical) magnetic field, B{\mathbf B}B, which is confined in a spatial region from which the magnetic field cannot escape. Even though the charged particle is not allowed to interact with the magnetic field, it accumulates a phase shift that affects the interference pattern produced. Not surprisingly, this apparent nonlocality is puzzling and counter intuitive. In this work, we provide an explanation that explains the physics underlying this apparent nonlocality. We find that the role of the confined magnetic field is to impart a puncture in the configuration space, R2\mathbb{R}^2R2, of the charge. Therefore, the quantum state corresponding to the charged quantum particle acquires the phase shift due to its response to the modified topology of the configuration space, R2−{0}\mathbb{R}^2-\{0\}R2−{0}, corresponding to the charge.