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arXiv:quant-ph/0207067 · Physical Review A

Free initial wave packets and the long-time behavior of the survival and nonescape probabilities

Manabu Miyamoto

Published Aug 7, 2003• 11 citations
arXivHTMLPDF

Abstract

The behavior of both the survival $S(t)$ and the nonescape $P(t)$ probabilities at long times for the one-dimensional free-particle system is shown to be closely connected to that of the initial wave packet at small momentum. We prove that both $S(t)$ and $P(t)$ asymptotically exhibit the same power-law decrease at long times, when the initial wave packet in momentum representation behaves as ${k}^{m}$ with $m=0$ or $1$ at small momentum. On the other hand, if the integer m becomes greater than $1,$ $S(t)$ and $P(t)$ decrease in different power laws at long times.

DOI
10.1103/physreva.68.022702

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The behavior of both the survival S(t)S(t)S(t) and the nonescape P(t)P(t)P(t) probabilities at long times for the one-dimensional free-particle system is shown to be closely connected to that of the initial wave packet at small momentum. We prove that both S(t)S(t)S(t) and P(t)P(t)P(t) asymptotically exhibit the same power-law decrease at long times, when the initial wave packet in momentum representation behaves as km{k}^{m}km with m=0m=0m=0 or 111 at small momentum. On the other hand, if the integer m becomes greater than 1,1,1, S(t)S(t)S(t) and P(t)P(t)P(t) decrease in different power laws at long times.
The power-law decrease of the survival probability S ( t ) at long times for the one-dimensional free-particle system is shown to be characterized by the small-momentum behavior of the initial wave packet. We consider and compare the result based on the resolvent analysis and that on the asymptotic method of the Fourier integral, where the effect of a specialty of a one-dimensional free resolvent and that of the discontinuity of initial states at zero momentum on the behavior of S ( t ) are pointed out.
The long time behaviour of the survival probability of the 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 such as t −2 m −1 for an arbitrary non-negative integer m as t → ∞, corresponding to the initial states with the condition ( k ) = O ( k m ) as k → 0.
The time evolution of quantum decay of an arbitrary state initially located within an interaction region of finite range is investigated. We compare the survival S(t)S(t)S(t) and nonescape P(t)P(t)P(t) probabilities. Our approach considers the full resonant spectra of the system using a novel representation of the time-dependent Green function. It is shown that, for an initial state near a resonance, S(t)S(t)S(t) and P(t)P(t)P(t) exhibit at long times a different behavior; not only the onset to a power law decay occurs at a different time, but also instead of the well known $S(t)\ensuremath{\sim}{t}^{\ensuremath{-}3}$ we obtain $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}1}$.
The long-time behavior of scattered wave packets $\ensuremath{\psi}(x,t)$ from a finite-range potential is investigated, by assuming $\ensuremath{\psi}(x,t)$ to be initially located outside the potential. It is then shown that $\ensuremath{\psi}(x,t)$ can asymptotically decrease in the various power laws at long time, according to its initial characteristics at small momentum. As an application, we consider the square-barrier potential system and demonstrate that $\ensuremath{\psi}(x,t)$ exhibits the asymptotic behavior ${t}^{\ensuremath{-}3∕2}$, while another behavior like ${t}^{\ensuremath{-}5∕2}$ can also appear for another $\ensuremath{\psi}(x,t)$.
A family of wave packets with power law tails are employed to analyze the long time dependence of the corresponding probability density. The densities, associated to packets for free particles in the one-dimensional space, with sufficiently long tails decay asymptotically in time following a t−βt^{-β}t−β law (with β<1β<1β<1), slower than the usual t−1t^{-1}t−1 one, including the particular case (lnt)2/t(ln t)^2/t(lnt)2/t not correctly discussed in F. Lillo and R. N. Mantegna (Phys. Rev. Lett. 84, 1061 (2000)).