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.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.

arXiv:quant-ph/0307020Physical Review AMay 30, 200238 citations

QUANTUM MECHANICAL COMPUTATION OF BILLIARD SYSTEMS WITH ARBITRARY SHAPES

İncı M. Erhan

Jan 1, 20030 citations

Time-dependent quantum circular billiard

Doniyor Babajanov, Davron U. Matrasulov, Zarifboy A. Sobirov +2 more

The motion of a quantum particle in a time-dependent circular billiard is studied on the basis of the Schrödinger equation with time-dependent boundary conditions.The cases of monotonically expanding (contracting), non-harmonically, harmonically breathing circles the case when billiard wall suddenly disappears are explored in detail.The exact analytical solutions for monotonically expanding and contracting circles are obtained.For all cases, the time-dependence of the quantum average energy is calculated.It is found that for an harmonically breathing circle, the average energy is time-periodic in the adiabatic regime with the same period as that of the oscillation.For intermediate frequencies which are comparable with the initial frequency of the particle in unperturbed billiard, such periodicity is broken.However, for very high frequencies, the average energy once again becomes periodic.A qualitative analysis of the border between adiabatic and non-adiabatic regimes is provided.The motion of a quantum particle in a time-dependent circular billiard is studied on the basis of the Schrödinger equation with time-dependent boundary conditions.The cases of monotonically expanding (contracting), non-harmonically, harmonically breathing circles the case when billiard wall suddenly disappears are explored in detail.The exact analytical solutions for monotonically expanding and contracting circles are obtained.For all cases, the time-dependence of the quantum average energy is calculated.It is found that for an harmonically breathing circle, the average energy is time-periodic in the adiabatic regime with the same period as that of the oscillation.For intermediate frequencies which are comparable with the initial frequency of the particle in unperturbed billiard, such periodicity is broken.However, for very high frequencies, the average energy once again becomes periodic.A qualitative analysis of the border between adiabatic and non-adiabatic regimes is provided.

Nanosystems Physics Chemistry MathematicsApr 16, 20152 citations

Classical limit of quantum mechanics and the quantum billiards problem

O. G. Mishnev

Russian Physics JournalMay 1, 19820 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=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.