Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:chao-dyn/9601010 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Wave chaos in quantum billiards with a small but finite-size scatterer

Takaomi Shigehara, Taksu Cheon

Published Aug 1, 1996• 26 citations
arXivHTMLPDF

Abstract

We study the low-energy quantum spectra of two-dimensional rectangular billiards with a small but finite-size scatterer inside. We start by examining the spectral properties of billiards with a single pointlike scatterer. The problem is formulated in terms of the self-adjoint extension theory of functional analysis. The condition for the appearance of so-called wave chaos is clarified. We then relate the pointlike scatterer to a finite-size scatterer through an appropriate truncation of the basis. We show that the signature of wave chaos in low-energy states is most prominent when the scatterer is weakly attractive. As an illustration, numerical results of a rectangular billiard with a small rectangular scatterer inside are exhibited. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreve.54.1321

Related papers

Chaos Induced by Quantization

Takaomi Shigehara, Hiroshi Mizoguchi, Taketoshi Mishima +1 more

In this paper, we show that two-dimensional billiards with point interactions inside exhibit a chaotic nature in the microscopic world, although their classical counterpart is non-chaotic. After deriving the transition matrix of the system by using the self-adjoint extension theory of functional analysis, we deduce the general condition for the appearance of chaos. The prediction is confirmed by numerically examining the statistical properties of energy spectrum of rectangular billiards with multiple point interactions inside. The dependence of the level statistics on the strength as well as the number of the scatterers is displayed. KEYWORDS: wave chaos, quantum mechanics, pseudointegrable billiard, point interaction, functional analysis

arXiv:quant-ph/9802008•IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences•Feb 3, 1998•6 citations

Geometric Phase in Quantum Billiards with a Pointlike Scatterer

Taksu Cheon, Takaomi Shigehara

We examine the quantum energy levels of rectangular billiards with a pointlike scatterer in one and two dimensions. By varying the location and the strength of the scatterer, we systematically find diabolical degeneracies among various levels. The associated Berry phase is illustrated, and the existence of localized wave functions is pointed out. In one dimension, even the ground state is shown to display the sign reversal with a mechanism to circumvent the Sturm-Liouville theorem.

arXiv:hep-th/9512164•Physical Review Letters•Mar 11, 1996•11 citations

Expansion method for stationary states of quantum billiards

David L. Kaufman, Ioan Kosztin, Klaus J. Schulten

A simple expansion method for numerically calculating the energy levels and the corresponding wave functions of a quantum particle in a two-dimensional infinite potential well with arbitrary shape (quantum billiard) is presented. The method permits the study of quantum billiards in an introductory quantum mechanics course. According to the method, wave functions inside the billiard are expressed in terms of an expansion of a complete set of orthonormal functions defined in a surrounding rectangle for which the Dirichlet boundary conditions apply, while approximating the billiard boundary by a potential energy step of a sufficiently large size. Numerical implementations of the method are described and applied to determine the energies and wave functions for quarter-circle, circle, and triangle billiards. Finally, the expansion method is applied to investigate the quantum signatures of chaos in a classically chaotic generic-triangle billiard.

American Journal of Physics•Feb 1, 1999•56 citations

Quantum spectra and classical periodic orbits in a rectangular billiard

Chuan‐Lu Yang

By using the representation of SU(2) coherent states,we construct the wave functions related to the classical periodic orbits in the 2D rectangular billiard.The relation between the classical periodic orbits and the quantum wave functions provide a more physical description of a phenomenon than the true eigenstates in mesoscopic systems.Furthermore,by using the periodic orbit theory,we calculate the Fourier transform ρ(L) of the quantum spectra of the two-dimensional rectangular billiard system.The resulting of the peaks in plots of |ρN(L)|2 versus L are compared to the lengths of the classical periodic orbits of the particle moving in the 2D rectangular cavity.The agreement of the peaks between |ρN(L)|2 and the length of the classical periodic orbit suggests the correspondence of quantum mechanics and classical mechanics.

Journal of Atomic and Molecular Physics•Jan 1, 2007•0 citations

The quantum spectral analysis of the two-dimensional annular billiard system

Yanhui Zhang, Zhang Ji-Quan, Xu Xue-You +1 more

Based on the extended closed-orbit theory together with spectral analysis, this paper studies the correspondence between quantum mechanics and the classical counterpart in a two-dimensional annular billiard. The results demonstrate that the Fourier-transformed quantum spectra are in very good accordance with the lengths of the classical ballistic trajectories, whereas spectral strength is intimately associated with the shapes of possible open orbits connecting arbitrary two points in the annular cavity. This approach facilitates an intuitive understanding of basic quantum features such as quantum interference, locations of the wavefunctions, and allows quantitative calculations in the range of high energies, where full quantum calculations may become impractical in general. This treatment provides a thread to explore the properties of microjunction transport and even quantum chaos under the much more general system.

Chinese Physics B•Jan 1, 2009•2 citations

Constraint operator solution to quantum billiard problems

David A. McGrew, W. Bauer

We introduce an additional method to solve Schr\"odinger's equation for a free particle in an infinite well of arbitrary shape (the Helmholtz equation with Dirichlet boundary conditions), a problem of interest in the area of quantum chaos. We expand the wave function in a basis of products of sine functions, then use the constraint operator to contain the wave function to a region within the domain of the basis functions. In this manner, a quantum billiard problem of arbitrary shape can be solved. Several methods exist to solve problems of this sort, but as recent work reviewing these methods has shown, all have shortcomings. Our work represents a different direction in the solution of these problems. Our method is different in that it provides a means of computing an eigenbasis. It is also interesting from a physical standpoint in that it can represent the Hamiltonian of a classically chaotic system in the basis of a classically regular system. \textcopyright{} 1996 The American Physical Society.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Nov 1, 1996•4 citations
We study the low-energy quantum spectra of two-dimensional rectangular billiards with a small but finite-size scatterer inside. We start by examining the spectral properties of billiards with a single pointlike scatterer. The problem is formulated in terms of the self-adjoint extension theory of functional analysis. The condition for the appearance of so-called wave chaos is clarified. We then relate the pointlike scatterer to a finite-size scatterer through an appropriate truncation of the basis. We show that the signature of wave chaos in low-energy states is most prominent when the scatterer is weakly attractive. As an illustration, numerical results of a rectangular billiard with a small rectangular scatterer inside are exhibited. \textcopyright{} 1996 The American Physical Society.
In this paper, we show that two-dimensional billiards with point interactions inside exhibit a chaotic nature in the microscopic world, although their classical counterpart is non-chaotic. After deriving the transition matrix of the system by using the self-adjoint extension theory of functional analysis, we deduce the general condition for the appearance of chaos. The prediction is confirmed by numerically examining the statistical properties of energy spectrum of rectangular billiards with multiple point interactions inside. The dependence of the level statistics on the strength as well as the number of the scatterers is displayed. KEYWORDS: wave chaos, quantum mechanics, pseudointegrable billiard, point interaction, functional analysis
A simple expansion method for numerically calculating the energy levels and the corresponding wave functions of a quantum particle in a two-dimensional infinite potential well with arbitrary shape (quantum billiard) is presented. The method permits the study of quantum billiards in an introductory quantum mechanics course. According to the method, wave functions inside the billiard are expressed in terms of an expansion of a complete set of orthonormal functions defined in a surrounding rectangle for which the Dirichlet boundary conditions apply, while approximating the billiard boundary by a potential energy step of a sufficiently large size. Numerical implementations of the method are described and applied to determine the energies and wave functions for quarter-circle, circle, and triangle billiards. Finally, the expansion method is applied to investigate the quantum signatures of chaos in a classically chaotic generic-triangle billiard.
By using the representation of SU(2) coherent states,we construct the wave functions related to the classical periodic orbits in the 2D rectangular billiard.The relation between the classical periodic orbits and the quantum wave functions provide a more physical description of a phenomenon than the true eigenstates in mesoscopic systems.Furthermore,by using the periodic orbit theory,we calculate the Fourier transform ρ(L) of the quantum spectra of the two-dimensional rectangular billiard system.The resulting of the peaks in plots of |ρN(L)|2 versus L are compared to the lengths of the classical periodic orbits of the particle moving in the 2D rectangular cavity.The agreement of the peaks between |ρN(L)|2 and the length of the classical periodic orbit suggests the correspondence of quantum mechanics and classical mechanics.
Based on the extended closed-orbit theory together with spectral analysis, this paper studies the correspondence between quantum mechanics and the classical counterpart in a two-dimensional annular billiard. The results demonstrate that the Fourier-transformed quantum spectra are in very good accordance with the lengths of the classical ballistic trajectories, whereas spectral strength is intimately associated with the shapes of possible open orbits connecting arbitrary two points in the annular cavity. This approach facilitates an intuitive understanding of basic quantum features such as quantum interference, locations of the wavefunctions, and allows quantitative calculations in the range of high energies, where full quantum calculations may become impractical in general. This treatment provides a thread to explore the properties of microjunction transport and even quantum chaos under the much more general system.
We introduce an additional method to solve Schr\"odinger's equation for a free particle in an infinite well of arbitrary shape (the Helmholtz equation with Dirichlet boundary conditions), a problem of interest in the area of quantum chaos. We expand the wave function in a basis of products of sine functions, then use the constraint operator to contain the wave function to a region within the domain of the basis functions. In this manner, a quantum billiard problem of arbitrary shape can be solved. Several methods exist to solve problems of this sort, but as recent work reviewing these methods has shown, all have shortcomings. Our work represents a different direction in the solution of these problems. Our method is different in that it provides a means of computing an eigenbasis. It is also interesting from a physical standpoint in that it can represent the Hamiltonian of a classically chaotic system in the basis of a classically regular system. \textcopyright{} 1996 The American Physical Society.