Quantum wave-packet revivals in circular billiards

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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 [Formula Presented] states (central wave packets with no momentum) the revival time is [Formula Presented] where [Formula Presented] is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that [Formula Presented] and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of [Formula Presented] 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.

Original languageEnglish (US)
Number of pages1
JournalPhysical Review A - Atomic, Molecular, and Optical Physics
Volume65
Issue number6
DOIs
StatePublished - Jan 1 2002

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wave packets
angular momentum
momentum
triangles
time dependence
periodic variations
energy spectra
eigenvalues
kinetic energy
orbits
geometry
energy

All Science Journal Classification (ASJC) codes

  • Atomic and Molecular Physics, and Optics

Cite this

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title = "Quantum wave-packet revivals in circular billiards",
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 [Formula Presented] states (central wave packets with no momentum) the revival time is [Formula Presented] where [Formula Presented] is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that [Formula Presented] and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of [Formula Presented] 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.",
author = "Robinett, {Richard Wallace} and Steven Heppelmann",
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T1 - Quantum wave-packet revivals in circular billiards

AU - Robinett, Richard Wallace

AU - Heppelmann, Steven

PY - 2002/1/1

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N2 - 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 [Formula Presented] states (central wave packets with no momentum) the revival time is [Formula Presented] where [Formula Presented] is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that [Formula Presented] and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of [Formula Presented] 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.

AB - 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 [Formula Presented] states (central wave packets with no momentum) the revival time is [Formula Presented] where [Formula Presented] is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that [Formula Presented] and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of [Formula Presented] 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.

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