Physics – Quantum Physics
Scientific paper
2002-07-11
J. Phys. A 35, 7159 (2002).
Physics
Quantum Physics
15 pages, to appear in J. Phys. A
Scientific paper
10.1088/0305-4470/35/33/312
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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