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

Journal of Physics A Mathematical and General

Level crossings and commuting observables for the quantum elliptic billiard

Ariel J.S. Traiber, A. J. Fendrik, M. Bernath

Published May 7, 1989• 19 citations
Publisher

Abstract

The energy spectrum of the quantum elliptic billiard is obtained by solving the Schrodinger equation in elliptic coordinates. The corresponding eigenstates also diagonalise an operator B which commutes with H. A numerical search of the exact eigenvalues of B and H permits one to follow each state as a function of the deformation parameter mu . Geometrical arguments, also valid for the simpler problem of the rectangular box, allow one to understand the results obtained.

DOI
10.1088/0305-4470/22/9/003

Related papers

The elliptic billiard: subtleties of separability

Ramses van Zon, Th.W. Ruijgrok

Some of the subtleties of the integrability of the elliptic quantum billiard are discussed. Considering a well known classical constant of the motion in the quantum case, we find that a naive calculation of the commutator with the Hamiltonian does not show whether or not it is zero. It is shown how this problem can be solved. A geometric picture is given that reveals why levels of a separable system cross. It is shown that the repulsions found by Ayant and Arvieu are computational effects and that the method used by Traiber et al is related to the present picture which explains the crossings they find. An asymptotic formula for the energy levels is derived and it is found that the statistical quantities of the spectrum P ( s ) and have the form expected for an integrable system.

arXiv:chao-dyn/9702006•European Journal of Physics•Jan 1, 1998•8 citations

Classical and quantum description of the two-dimensional simple harmonic oscillator in elliptic coordinates

A. J. Fendrik, M. Bernath

The Schr\"odinger equation for the two-dimensional simple harmonic oscillator is solved using elliptic coordinates where it is separable. The separability of the problem in such coordinates is independent of the selection of the focal distance. The solutions are labeled by the total number of quanta N and by a set of characteristic values b corresponding to the eigenvalues of an observable B^, which does not commute with L^, the total angular momentum or H${^}_{x}$, the energy associated with the x degree of freedom. The well-known quantum energies as well as the characteristic values are obtained by imposing physical polynomial solutions.

Physical Review A•Oct 1, 1989•5 citations

From quantum spectra to classical orbits: Varying rectangular billiards

Jun Lu

The quantum-spectrum function is introduced by means of the eigenvalues and the eigenfunctions in the system of varying rectangular billiards. It is found that the Fourier transform of the quantum-spectrum function corresponds with the classical orbits in the system. These results give new evidence about the classical-quantum correspondence.

Progress in Natural Science Materials International•Jun 17, 2008•4 citations

The Elliptic Billiard: Subtleties of Separability

Ramses van Zon, Th.W. Ruijgrok

Some of the subtleties of the integrability of the elliptic quantum billiard are discussed. A well known classical constant of the motion has in the quantum case an ill-defined commutator with the Hamiltonian. It is shown how this problem can be solved. A geometric picture is given revealing why levels of a separable system cross. It is shown that the repulsions found by Ayant and Arvieu are computational effects and that the method used by Traiber et al is related to the present picture which explains the crossings they find. An asymptotic formula for the energy-levels is derived and it is found that the statistical quantities of the spectrum P(s) and ¯∆(L) have the form expected for an integrable system. Quelques-unes des subtilités de l’intégrabilité du billard elliptique quantique sont discutées. Dans le cas quantique, le commutateur avec l’Hamiltonien d’une certaine quantité connue pour ˜etre une constante du mouvement en méchanique classique, s’avère mal défini. On démontre ici comment on peut résoudre ce problème. Pour expliquer que les niveaux d’énergie d’un sytème séparable se croisent, une représentation géométrique est utilisée. On montre que les répulsions trouvées par Ayant et Arvieu sont des effets numériques et que la méthode employée par Traiber et al est liée à la représentation géométrique considérée ici, ce qui explique qu’ils trouvent aussi un croisement des niveaux d’énergie. Une expression asymptotique est derivée pour les niveaux d’énergie et les quantités statistiques du spectre P(s) et ¯ ∆(L) sont obtenues. Ils ont la forme prédite pour des systèmes intégrables.

•Jan 1, 1997•5 citations

Wave packet construction in two-dimensional quantum billiards: Blueprints for the square, equilateral triangle, and circular cases

M. A. Doncheski, S. Heppelmann, R. W. Robinett +1 more

We present quasianalytical and numerical calculations of Gaussian wave packet solutions of the Schrödinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and N→∞ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wave functions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wave packets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasiclassical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.

arXiv:quant-ph/0307070•American Journal of Physics•May 12, 2003•33 citations

Elliptic Quantum Billiard

Holger Waalkens, Jan Wiersig, Holger R. Dullin

The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phase space into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schrödinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the Berry-Tabor trace formula, resulting in a summation over non-periodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the The semiclassical quantization of a Hamiltonian system is deeply connected to the structure of its phase space. The generic Hamiltonian system contains a complicated mixture of near-integrable and

•Jan 1, 1997•40 citations
The energy spectrum of the quantum elliptic billiard is obtained by solving the Schrodinger equation in elliptic coordinates. The corresponding eigenstates also diagonalise an operator B which commutes with H. A numerical search of the exact eigenvalues of B and H permits one to follow each state as a function of the deformation parameter mu . Geometrical arguments, also valid for the simpler problem of the rectangular box, allow one to understand the results obtained.
Some of the subtleties of the integrability of the elliptic quantum billiard are discussed. Considering a well known classical constant of the motion in the quantum case, we find that a naive calculation of the commutator with the Hamiltonian does not show whether or not it is zero. It is shown how this problem can be solved. A geometric picture is given that reveals why levels of a separable system cross. It is shown that the repulsions found by Ayant and Arvieu are computational effects and that the method used by Traiber et al is related to the present picture which explains the crossings they find. An asymptotic formula for the energy levels is derived and it is found that the statistical quantities of the spectrum P ( s ) and have the form expected for an integrable system.
The Schr\"odinger equation for the two-dimensional simple harmonic oscillator is solved using elliptic coordinates where it is separable. The separability of the problem in such coordinates is independent of the selection of the focal distance. The solutions are labeled by the total number of quanta N and by a set of characteristic values b corresponding to the eigenvalues of an observable B^, which does not commute with L^, the total angular momentum or H${^}_{x}$, the energy associated with the x degree of freedom. The well-known quantum energies as well as the characteristic values are obtained by imposing physical polynomial solutions.
Some of the subtleties of the integrability of the elliptic quantum billiard are discussed. A well known classical constant of the motion has in the quantum case an ill-defined commutator with the Hamiltonian. It is shown how this problem can be solved. A geometric picture is given revealing why levels of a separable system cross. It is shown that the repulsions found by Ayant and Arvieu are computational effects and that the method used by Traiber et al is related to the present picture which explains the crossings they find. An asymptotic formula for the energy-levels is derived and it is found that the statistical quantities of the spectrum P(s) and ¯∆(L) have the form expected for an integrable system. Quelques-unes des subtilités de l’intégrabilité du billard elliptique quantique sont discutées. Dans le cas quantique, le commutateur avec l’Hamiltonien d’une certaine quantité connue pour ˜etre une constante du mouvement en méchanique classique, s’avère mal défini. On démontre ici comment on peut résoudre ce problème. Pour expliquer que les niveaux d’énergie d’un sytème séparable se croisent, une représentation géométrique est utilisée. On montre que les répulsions trouvées par Ayant et Arvieu sont des effets numériques et que la méthode employée par Traiber et al est liée à la représentation géométrique considérée ici, ce qui explique qu’ils trouvent aussi un croisement des niveaux d’énergie. Une expression asymptotique est derivée pour les niveaux d’énergie et les quantités statistiques du spectre P(s) et ¯ ∆(L) sont obtenues. Ils ont la forme prédite pour des systèmes intégrables.
We present quasianalytical and numerical calculations of Gaussian wave packet solutions of the Schrödinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and N→∞ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wave functions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wave packets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasiclassical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.
The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phase space into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schrödinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the Berry-Tabor trace formula, resulting in a summation over non-periodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the The semiclassical quantization of a Hamiltonian system is deeply connected to the structure of its phase space. The generic Hamiltonian system contains a complicated mixture of near-integrable and