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:quant-ph/0307020 · Physical Review A

Quantum wave-packet revivals in circular billiards

Richard W. Robinett, S. Heppelmann

Published May 30, 2002• 38 citations
arXivHTMLPDF

Abstract

We examine the long-term time dependence of Gaussian wave packets in a circular infinite well (billiard) system and find that there are approximate revivals. For the special case of purely $m=0$ states (central wave packets with no momentum) the revival time is ${T}_{\mathrm{rev}}^{(m=0)}=8\ensuremath{\mu}{R}^{2}/\ensuremath{\Elzxh}\ensuremath{\pi},$ where $\ensuremath{\mu}$ is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that ${T}_{\mathrm{rev}}^{(m\ensuremath{\ne}0)}=({\ensuremath{\pi}}^{2}{/2)T}_{\mathrm{rev}}^{(m=0)}\ensuremath{\approx}{5T}_{\mathrm{rev}}^{(m=0)}$ and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of $m\ensuremath{\ne}0$ states is increased. The dependence of the revival structure on the initial position, energy, and angular momentum of the wave packet and the connection to the energy spectrum is discussed in detail. The results are also compared to two other highly symmetrical two-dimensional infinite well geometries with exact revivals, namely, the square and equilateral triangle billiards. We also show explicitly how the classical periodicity for closed orbits in a circular billiard arises from the energy eigenvalue spectrum, using a WKB analysis.

DOI
10.1103/physreva.65.062103

Related papers

Quantum mechanical analysis of the equilateral triangle billiard: periodic orbit theory and wave packet revivals

M. A. Doncheski, Richard W. Robinett

Using the fact that the energy eigenstates of the equilateral triangle infinite well (or billiard) are available in closed form, we examine the connections between the energy eigenvalue spectrum and the classical closed paths in this geometry, using both periodic orbit theory and the short-term semi-classical behavior of wave packets. We also discuss wave packet revivals and show that there are exact revivals, for all wave packets, at times given by $T_{rev} = 9 mu a^2/4hbar pi$ where $a$ and $mu$ are the length of one side and the mass of the point particle respectively. We find additional cases of exact revivals with shorter revival times for zero-momentum wave packets initially located at special symmetry points inside the billiard. Finally, we discuss simple variations on the equilateral ($60^{circ}-60^{circ}-60^{circ}$) triangle, such as the half equilateral ($30^{circ}-60^{circ}-90^{circ}$) triangle and other `foldings', which have related energy spectra and revival structures.

•Jul 8, 2003•16 citations

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

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

We present quasi-analytical and numerical calculations of Gaussian wave packet solutions of the Schr\\"odinger 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 $N \\to \\infty$ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions 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 wavepackets 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, quasi-classical 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.

•Jul 9, 2003•20 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

Quantum wave packet analysis in the rectangular billiard

Lin Sheng-Lu

The use of wave packet to analyze the dynamics of quantum mechanical systems is an increasingly important method to the study of the classical-quantum correspondence.Using the quantum Gaussian wave packet analysis method,we calculate the autocorrelation function of the rectangular billiard,the peak positions of the autocorrelation function match well with the periods of the classical periodic orbits,which show that the period of the classical orbits can be produced by the time-dependent quantum wave packet method.We also discuss wave packet revivals and fractional revivals in the rectangular billiard,the results show that there are exact revival for all wave packet at each revival time.We find additional cases of exact revivals with short revival times for zero-momentum wave packets initially located at special symmetry point inside the billiard.

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

Improving wave-packet revivals in circular billiards by applying constant magnetic fields

Guilherme Jurkevicz Delben, André Gusso, M. G. E. da Luz

We show that the revivals structure of Gaussian wave packets for a charged particle with nonzero linear momentum placed in a circular billiard can be considerably improved by applying a perpendicular constant magnetic field of correct chosen intensity. To obtain the desired enhancement, the field must be turned on only at certain time values. We also briefly investigate how the shape of the evoluted packet at the revivals, with and without an external $B$, compares with the initial state.

Physical Review A•May 31, 2006•1 citation

Quantum mechanics of the two-dimensional circular billiard plus baffle system and half-integral angular momentum

Richard W. Robinett

We examine the quantum mechanical eigensolutions of the two-dimensional infinite-well or quantum billiard system consisting of a circular boundary with an infinite barrier or baffle along a radius. Because of the change in boundary conditions, this system includes quantized angular momentum values corresponding to half-integral multiples of ℏ/2. We discuss the resulting energy eigenvalue spectrum and visualize some of the novel energy eigenstates found in this system. We also discuss the density of energy eigenvalues, N ( E ), comparing this system to the standard circular well. These two billiard geometries have the same area ( A = π R 2 ), but different perimeters ( P = 2π R versus (2π + 2) R ), and we compare both cases to fits of N ( E ) which make use of purely geometric arguments involving only A and P . We also point out connections between the angular solutions of this system and the familiar pedagogical example of the one-dimensional infinite-well plus δ-function potential.

arXiv:quant-ph/0307035•European Journal of Physics•Mar 7, 2003•32 citations
We examine the long-term time dependence of Gaussian wave packets in a circular infinite well (billiard) system and find that there are approximate revivals. For the special case of purely m=0m=0m=0 states (central wave packets with no momentum) the revival time is Trev(m=0)=8\ensuremathμR2/\ensuremath\Elzxh\ensuremathπ,{T}_{\mathrm{rev}}^{(m=0)}=8\ensuremath{\mu}{R}^{2}/\ensuremath{\Elzxh}\ensuremath{\pi},Trev(m=0)​=8\ensuremathμR2/\ensuremath\Elzxh\ensuremathπ, where \ensuremathμ\ensuremath{\mu}\ensuremathμ is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that Trev(m\ensuremath≠0)=(\ensuremathπ2/2)Trev(m=0)\ensuremath≈5Trev(m=0){T}_{\mathrm{rev}}^{(m\ensuremath{\ne}0)}=({\ensuremath{\pi}}^{2}{/2)T}_{\mathrm{rev}}^{(m=0)}\ensuremath{\approx}{5T}_{\mathrm{rev}}^{(m=0)}Trev(m\ensuremath=0)​=(\ensuremathπ2/2)Trev(m=0)​\ensuremath≈5Trev(m=0)​ and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of m\ensuremath≠0m\ensuremath{\ne}0m\ensuremath=0 states is increased. The dependence of the revival structure on the initial position, energy, and angular momentum of the wave packet and the connection to the energy spectrum is discussed in detail. The results are also compared to two other highly symmetrical two-dimensional infinite well geometries with exact revivals, namely, the square and equilateral triangle billiards. We also show explicitly how the classical periodicity for closed orbits in a circular billiard arises from the energy eigenvalue spectrum, using a WKB analysis.
Using the fact that the energy eigenstates of the equilateral triangle infinite well (or billiard) are available in closed form, we examine the connections between the energy eigenvalue spectrum and the classical closed paths in this geometry, using both periodic orbit theory and the short-term semi-classical behavior of wave packets. We also discuss wave packet revivals and show that there are exact revivals, for all wave packets, at times given by Trev=9mua2/4hbarpiT_{rev} = 9 mu a^2/4hbar piTrev​=9mua2/4hbarpi where aaa and mumumu are the length of one side and the mass of the point particle respectively. We find additional cases of exact revivals with shorter revival times for zero-momentum wave packets initially located at special symmetry points inside the billiard. Finally, we discuss simple variations on the equilateral (60circ−60circ−60circ60^{circ}-60^{circ}-60^{circ}60circ−60circ−60circ) triangle, such as the half equilateral (30circ−60circ−90circ30^{circ}-60^{circ}-90^{circ}30circ−60circ−90circ) triangle and other `foldings', which have related energy spectra and revival structures.
We present quasi-analytical and numerical calculations of Gaussian wave packet solutions of the Schr\\"odinger 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 NtoinftyN \\to \\inftyNtoinfty regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions 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 wavepackets 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, quasi-classical 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.
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 use of wave packet to analyze the dynamics of quantum mechanical systems is an increasingly important method to the study of the classical-quantum correspondence.Using the quantum Gaussian wave packet analysis method,we calculate the autocorrelation function of the rectangular billiard,the peak positions of the autocorrelation function match well with the periods of the classical periodic orbits,which show that the period of the classical orbits can be produced by the time-dependent quantum wave packet method.We also discuss wave packet revivals and fractional revivals in the rectangular billiard,the results show that there are exact revival for all wave packet at each revival time.We find additional cases of exact revivals with short revival times for zero-momentum wave packets initially located at special symmetry point inside the billiard.
We show that the revivals structure of Gaussian wave packets for a charged particle with nonzero linear momentum placed in a circular billiard can be considerably improved by applying a perpendicular constant magnetic field of correct chosen intensity. To obtain the desired enhancement, the field must be turned on only at certain time values. We also briefly investigate how the shape of the evoluted packet at the revivals, with and without an external BBB, compares with the initial state.
We examine the quantum mechanical eigensolutions of the two-dimensional infinite-well or quantum billiard system consisting of a circular boundary with an infinite barrier or baffle along a radius. Because of the change in boundary conditions, this system includes quantized angular momentum values corresponding to half-integral multiples of ℏ/2. We discuss the resulting energy eigenvalue spectrum and visualize some of the novel energy eigenstates found in this system. We also discuss the density of energy eigenvalues, N ( E ), comparing this system to the standard circular well. These two billiard geometries have the same area ( A = π R 2 ), but different perimeters ( P = 2π R versus (2π + 2) R ), and we compare both cases to fits of N ( E ) which make use of purely geometric arguments involving only A and P . We also point out connections between the angular solutions of this system and the familiar pedagogical example of the one-dimensional infinite-well plus δ-function potential.