dorsal/arxiv
View SchemaQuantum wave packet revivals in circular billiards
| Authors | R. W. Robinett, S. Heppelmann |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0307020 |
| URL | https://arxiv.org/abs/quant-ph/0307020 |
| DOI | 10.1103/PhysRevA.65.062103 |
| Journal | Phys. Rev. A 65 (2002) 062103 |
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 $T_{rev}^{(m=0)} = 8\mu R^2/\hbar \pi$, where $\mu$ is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that $T_{rev}^{(m \neq 0)} = (\pi^2/2) T_{rev}^{(m=0)} \approx 5T_{rev}^{(m=0)}$ and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of $m \neq 0$ states is/are 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 2D 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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"abstract": "We examine the long-term time-dependence of Gaussian wave packets in a\ncircular infinite well (billiard) system and find that there are approximate\nrevivals. For the special case of purely $m=0$ states (central wave packets\nwith no momentum) the revival time is $T_{rev}^{(m=0)} = 8\\mu R^2/\\hbar \\pi$,\nwhere $\\mu$ is the mass of the particle, and the revivals are almost exact. For\nall other wave packets, we find that $T_{rev}^{(m \\neq 0)} = (\\pi^2/2)\nT_{rev}^{(m=0)} \\approx 5T_{rev}^{(m=0)}$ and the nature of the revivals\nbecomes increasingly approximate as the average angular momentum or number of\n$m \\neq 0$ states is/are increased. The dependence of the revival structure on\nthe initial position, energy, and angular momentum of the wave packet and the\nconnection to the energy spectrum is discussed in detail. The results are also\ncompared to two other highly symmetrical 2D infinite well geometries with exact\nrevivals, namely the square and equilateral triangle billiards. We also show\nexplicitly how the classical periodicity for closed orbits in a circular\nbilliard arises from the energy eigenvalue spectrum, using a WKB analysis.",
"arxiv_id": "quant-ph/0307020",
"authors": [
"R. W. Robinett",
"S. Heppelmann"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.65.062103",
"journal_ref": "Phys. Rev. A 65 (2002) 062103",
"title": "Quantum wave packet revivals in circular billiards",
"url": "https://arxiv.org/abs/quant-ph/0307020"
},
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