dorsal/arxiv
View SchemaThe various power decays of the survival probability at long times for free quantum particle
| Authors | Manabu Miyamoto |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0207066 |
| URL | https://arxiv.org/abs/quant-ph/0207066 |
| DOI | 10.1088/0305-4470/35/33/312 |
| Journal | J. Phys. A 35, 7159 (2002). |
Abstract
The long time behaviour of the survival probability of initial state and its dependence on the initial states are considered, for the one dimensional free quantum particle. We derive the asymptotic expansion of the time evolution operator at long times, in terms of the integral operators. This enables us to obtain the asymptotic formula for the survival probability of the initial state $\psi (x)$, which is assumed to decrease sufficiently rapidly at large $|x|$. We then show that the behaviour of the survival probability at long times is determined by that of the initial state $\psi$ at zero momentum $k=0$. Indeed, it is proved that the survival probability can exhibit the various power-decays like $t^{-2m-1}$ for an arbitrary non-negative integers $m$ as $t \to \infty $, corresponding to the initial states with the condition $\hat{\psi} (k) = O(k^m)$ as $k\to 0$.
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"abstract": "The long time behaviour of the survival probability of initial state and its\ndependence on the initial states are considered, for the one dimensional free\nquantum particle. We derive the asymptotic expansion of the time evolution\noperator at long times, in terms of the integral operators. This enables us to\nobtain the asymptotic formula for the survival probability of the initial state\n$\\psi (x)$, which is assumed to decrease sufficiently rapidly at large $|x|$.\nWe then show that the behaviour of the survival probability at long times is\ndetermined by that of the initial state $\\psi$ at zero momentum $k=0$. Indeed,\nit is proved that the survival probability can exhibit the various power-decays\nlike $t^{-2m-1}$ for an arbitrary non-negative integers $m$ as $t \\to \\infty $,\ncorresponding to the initial states with the condition $\\hat{\\psi} (k) =\nO(k^m)$ as $k\\to 0$.",
"arxiv_id": "quant-ph/0207066",
"authors": [
"Manabu Miyamoto"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/35/33/312",
"journal_ref": "J. Phys. A 35, 7159 (2002).",
"title": "The various power decays of the survival probability at long times for free quantum particle",
"url": "https://arxiv.org/abs/quant-ph/0207066"
},
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