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arXiv quant-ph/0207066quant-ph
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自由量子粒子长时间生存概率的各种幂次衰减

The various power decays of the survival probability at long times for free quantum particle

Manabu Miyamoto

备注：
15 pages, to appear in J. Phys. A
发表信息：
J. Phys. A 35, 7159 (2002).
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摘要：

研究了一维自由量子粒子初态生存概率的长时间行为及其对初态的依赖性。我们得到了时间演化算子在长时间内的积分算子的渐近展开式。这使我们能够获得初始状态$ψ(x)$的生存概率的渐近公式，假设它在大$|x|$时下降得足够快。然后，我们表明，在很长一段时间的生存概率的行为是由零动量$k=0$的初始状态$ψ$。事实上，它被证明，生存概率可以表现出各种功率衰减$t^{-2m-1}$的任意非负整数$m$作为$t \to \infty $，对应于条件$\hatψ (k) = O(k^m)$作为$k\to 0$的初始状态。

英文摘要：

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 $ψ(x)$, 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 $ψ$ 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 $\hatψ (k) = O(k^m)$ as $k\to 0$.

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