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:1709.03650 · Reviews of Modern Physics

Nodal portraits of quantum billiards: Domains, lines, and statistics

Sudhir R. Jain, Rhine Samajdar

Published Dec 26, 2017• 46 citations
arXivHTMLPDF

Abstract

The 200-year old studies by Chladni, of the sand figures of vibrating plates, live on today in the investigation of the wave mechanics of nonintegrable systems. This review gives an in-depth survey of the contemporary mathematical analysis of the wave-function statistics of the billiard problem, uncovering the manifestations of classical chaos in the wave-mechanical world. The extensive experiments of recent years are surveyed. The illustrations show the continuing Chladnian splendor of the current-day subject.

DOI
10.1103/revmodphys.89.045005

Related papers

Microwave studies in irregularly shaped billiards

Giulio Casati, Boris V. Chirikov

Until a short time ago, the study of the quantum-mechanical behaviour of classically chaotic systems was an exclusive domain of the theoretical physicist. Model systems amongst others were irregularly shaped billiards, such as the Sinai and the stadium billiard. An alternative experimental approach to study these systems has recently been demonstrated using the fact that the time-independent Schrödinger and wave equations are mathematically equivalent (though, in general, the boundary conditions are different). This chapter concentrates on the presentation of microwave studies of billiards, but experiments with vibrating plates and water surfaces waves are also briefly discussed. Topics are the statistical properties of spectra, level dynamics with respect to geometrical changes of the billiards, and the close connection between quantum-mechanical spectra and classical trajectories. Introduction One day in February 1808 Ernst F. Chladni who was then on a long stay in Paris received an invitation to the Tuileries to give a demonstration of his famous experiments with vibrating plates in the presence of Napoleon and an illustrious audience. The Emperor was impressed by the performance and offered a prize of 3000 francs for the correct mathematical explanation of the sound figures which was still lacking at that time. The prize was paid in 1816 to the French mathematician Sophie Germain, in spite of the fact that her solution was still incomplete. The correct explanation for circular plates was not given until 1850 by Robert Kirchhoff (all the above details are taken from the book of F. Melde on the life of Chladni [1]).

Cambridge University Press eBooks•Apr 27, 1995•2 citations

Billiard experiments

H.‐J. Stöckmann

Until about 1990 only a very small number of experiments on the quantum mechanics of chaotic systems existed, apart from the early studies of nuclear spectra [Por65]. In this context the experiments on hydrogen atoms in strong microwave fields by Bayfield and Koch [Bay74], and in strong magnetic fields by Welge and his group [Hol86, Mai86] have to be mentioned in particular. The studies of irregularly shaped microwave cavities by Stöckmann and Stein [Stö90] have introduced a new type of quantum chaos research. The microwave billiards and a number of variants to be discussed in this chapter are analogue systems, as they use the equivalence of the Helmholtz equation and the time independent Schrödinger equation. Whether there is a complete correspondence with quantum mechanics or not depends on the respective boundary conditions. As most of the phenomena discussed in the following are common to all types of waves, this does not reduce the conclusiveness of the analogue experiments. Starting with a historical review, the state of the art in billiard experiments is presented with emphasis on a general survey and the technical background. The results and their quantum mechanical implications will be presented later. The hydrogen experiments, too, will be described in the proper context. The discussion of mesoscopic systems is restricted to billiard-like structures such as antidot lattices [Wei91], quantum dots [Mar92], and tunnelling devices [Fro94].

Cambridge University Press eBooks•Oct 13, 1999•0 citations

Systematically Constructing Mesoscopic Quantum States Relevant to Periodic Orbits in Integrable Billiards from Directionally Resolved Level Distributions

Y. F. Chen, Songqing Lin, Ru-Wei Chang +2 more

Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards.

Symmetry•Sep 22, 2023•2 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

The polygon quantum-billiard problem

Richard L. Liboff

The present work addresses the quantum polygon billiard problem with attention given to analytic and degeneracy properties of energy eigenstates. The ‘‘polygon ground-state theorem’’ is proven which states that the only polygons that contain respective ground states that are analytic in the closed domain of the entire polygon are the ‘‘elemental polygons’’ (defined in the text). The ‘‘polygon first excited-state theorem’’ is established which states that for every N-sided regular polygon, N equivalent first excited states exist, each of which contains a nodal curve that is a line of mirror symmetry of the related polygon. A vector description of nodal diagonal eigenstates is introduced to establish the second component of this theorem which indicates that the space of first excited states for the N-sided regular polygon is spanned by any two of these N nodal-diagonal eigenstates (i.e., the first excited state is twofold degenerate). At various levels of the discussion attention is drawn to the correspondence between the quantum and classical solutions for these configurations. Thus, for example, correspondence is demonstrated in the common source of integrability in the classical and quantum domains for three inclusive, nonoverlapping classes of polygons. Stemming from the preceding conclusions, a discussion is included on the possibility of employing transverse magnetic (TM) modes in a metal waveguide to determine if an equivalent quantum billiard configuration is chaotic.

Journal of Mathematical Physics•Feb 1, 1994•32 citations

Manifesting the evolution of eigenstates from quantum billiards to singular billiards in the strongly coupled limit with a truncated basis by usingRLCnetworks

P. H. Tuan, H. C. Liang, J. C. Tung +3 more

The coupling interaction between the driving source and the RLC network is explored and characterized as the effective impedance. The mathematical form of the derived effective impedance is verified to be identical to the meromorphic function of the singular billiards with a truncated basis. By using the derived impedance function, the resonant modes of the RLC network can be divided into the open-circuit and short-circuit states to manifest the evolution of eigenvalues and eigenstates from closed quantum billiards to the singular billiards with a truncated basis in the strongly coupled limit. The substantial differences of the wave patterns between the uncoupled and strongly coupled eigenmodes in the two-dimensional wave systems can be clearly revealed with the RLC network. Finally, the short-circuit resonant states are exploited to confirm that the experimental Chladni nodal-line patterns in the vibrating plate are the resonant modes subject to the strong coupling between the oscillation system and the driving source.

Physical Review E•Dec 7, 2015•7 citations
The 200-year old studies by Chladni, of the sand figures of vibrating plates, live on today in the investigation of the wave mechanics of nonintegrable systems. This review gives an in-depth survey of the contemporary mathematical analysis of the wave-function statistics of the billiard problem, uncovering the manifestations of classical chaos in the wave-mechanical world. The extensive experiments of recent years are surveyed. The illustrations show the continuing Chladnian splendor of the current-day subject.
Until a short time ago, the study of the quantum-mechanical behaviour of classically chaotic systems was an exclusive domain of the theoretical physicist. Model systems amongst others were irregularly shaped billiards, such as the Sinai and the stadium billiard. An alternative experimental approach to study these systems has recently been demonstrated using the fact that the time-independent Schrödinger and wave equations are mathematically equivalent (though, in general, the boundary conditions are different). This chapter concentrates on the presentation of microwave studies of billiards, but experiments with vibrating plates and water surfaces waves are also briefly discussed. Topics are the statistical properties of spectra, level dynamics with respect to geometrical changes of the billiards, and the close connection between quantum-mechanical spectra and classical trajectories. Introduction One day in February 1808 Ernst F. Chladni who was then on a long stay in Paris received an invitation to the Tuileries to give a demonstration of his famous experiments with vibrating plates in the presence of Napoleon and an illustrious audience. The Emperor was impressed by the performance and offered a prize of 3000 francs for the correct mathematical explanation of the sound figures which was still lacking at that time. The prize was paid in 1816 to the French mathematician Sophie Germain, in spite of the fact that her solution was still incomplete. The correct explanation for circular plates was not given until 1850 by Robert Kirchhoff (all the above details are taken from the book of F. Melde on the life of Chladni [1]).
Until about 1990 only a very small number of experiments on the quantum mechanics of chaotic systems existed, apart from the early studies of nuclear spectra [Por65]. In this context the experiments on hydrogen atoms in strong microwave fields by Bayfield and Koch [Bay74], and in strong magnetic fields by Welge and his group [Hol86, Mai86] have to be mentioned in particular. The studies of irregularly shaped microwave cavities by Stöckmann and Stein [Stö90] have introduced a new type of quantum chaos research. The microwave billiards and a number of variants to be discussed in this chapter are analogue systems, as they use the equivalence of the Helmholtz equation and the time independent Schrödinger equation. Whether there is a complete correspondence with quantum mechanics or not depends on the respective boundary conditions. As most of the phenomena discussed in the following are common to all types of waves, this does not reduce the conclusiveness of the analogue experiments. Starting with a historical review, the state of the art in billiard experiments is presented with emphasis on a general survey and the technical background. The results and their quantum mechanical implications will be presented later. The hydrogen experiments, too, will be described in the proper context. The discussion of mesoscopic systems is restricted to billiard-like structures such as antidot lattices [Wei91], quantum dots [Mar92], and tunnelling devices [Fro94].
Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards.
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.
The present work addresses the quantum polygon billiard problem with attention given to analytic and degeneracy properties of energy eigenstates. The ‘‘polygon ground-state theorem’’ is proven which states that the only polygons that contain respective ground states that are analytic in the closed domain of the entire polygon are the ‘‘elemental polygons’’ (defined in the text). The ‘‘polygon first excited-state theorem’’ is established which states that for every N-sided regular polygon, N equivalent first excited states exist, each of which contains a nodal curve that is a line of mirror symmetry of the related polygon. A vector description of nodal diagonal eigenstates is introduced to establish the second component of this theorem which indicates that the space of first excited states for the N-sided regular polygon is spanned by any two of these N nodal-diagonal eigenstates (i.e., the first excited state is twofold degenerate). At various levels of the discussion attention is drawn to the correspondence between the quantum and classical solutions for these configurations. Thus, for example, correspondence is demonstrated in the common source of integrability in the classical and quantum domains for three inclusive, nonoverlapping classes of polygons. Stemming from the preceding conclusions, a discussion is included on the possibility of employing transverse magnetic (TM) modes in a metal waveguide to determine if an equivalent quantum billiard configuration is chaotic.
The coupling interaction between the driving source and the RLC network is explored and characterized as the effective impedance. The mathematical form of the derived effective impedance is verified to be identical to the meromorphic function of the singular billiards with a truncated basis. By using the derived impedance function, the resonant modes of the RLC network can be divided into the open-circuit and short-circuit states to manifest the evolution of eigenvalues and eigenstates from closed quantum billiards to the singular billiards with a truncated basis in the strongly coupled limit. The substantial differences of the wave patterns between the uncoupled and strongly coupled eigenmodes in the two-dimensional wave systems can be clearly revealed with the RLC network. Finally, the short-circuit resonant states are exploited to confirm that the experimental Chladni nodal-line patterns in the vibrating plate are the resonant modes subject to the strong coupling between the oscillation system and the driving source.