arXiv:quant-ph/9804041 · Physical Review Letters

García-Calderón, Mateos, and Moshinsky Reply:

Gastón García‐Calderón, José L. Mateos, M. Moshińsky

Published May 11, 1998• 12 citations

Abstract

In our paper [Phys. Rev. Lett. 74, 337 (1995)], we derived an exact expression for the survival and nonescape probabilities as an expansion in terms of resonant states. It was shown that these quantities exhibit at long times a different behavior. Although both decay as a power law, they have different exponents. In this paper we show that, contrary to the claim in the Comment of R. M. Cavalcanti (quant-ph/9704023), the nonescape probability decay for long times as an inverse power law.

DOI: 10.1103/physrevlett.80.4354

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In [G. Garcia-Calderon, J. L. Mateos, and M. Moshinsky, Phys. Rev. Lett. 74, 337 (1995)], the time evolution of the quantum decay of a state initially located within an interaction region of finite range was investigated. In particular, it was shown that the survival and nonescape probabilities behave differently at very large times. The purpose of this Comment is to show that they have the same asymptotic behavior.

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arXiv:quant-ph/9804041 · Physical Review Letters

García-Calderón, Mateos, and Moshinsky Reply:

Gastón García‐Calderón, José L. Mateos, M. Moshińsky

Published May 11, 1998• 12 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.80.4354

Related papers

Comment on “Resonant Spectra and the Time Evolution of the Survival and Nonescape Probabilities”

R. M. Cavalcanti

arXiv:quant-ph/9704023Physical Review LettersMay 11, 199822 citations

Resonant Spectra and the Time Evolution of the Survival and Nonescape Probabilities

Gastón García‐Calderón, J. L. Mateos, M. Moshińsky

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 $S(t)$ and nonescape $P(t)$ 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, $S(t)$ and $P(t)$ 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 time evolution of quantum decay of an arbitrary state initially located within an interaction region of finite range is investigated. We compare the survival $S(t)$ and nonescape $P(t)$ 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, $S(t)$ and $P(t)$ 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 $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}1}$.

Physical Review LettersJan 16, 199587 citations

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Gastón García‐Calderón, I. K. A. ROMERO MALDONADO, Jorge Villavicencio

It is shown that a representation of the decaying wave function as a resonant sum plus a nonexponential integral term may be written as a purely discrete resonant sum by evaluating at long times the integral term by the steepest descents method, and then expanding the resulting expression in terms of resonant states. This leads to a representation that is valid along the exponential and the inverse power in time regimes. A model calculation using the $\ensuremath{\delta}$ potential allows us to make a comparison of the expansion with numerical integrations in terms of continuum wave functions and, in the long time regime, with an exact analytic expression of the integral term obtained using the steepest descents method. The results demonstrate that resonant states give a correct description of the long-time behavior of decay.It is shown that a representation of the decaying wave function as a resonant sum plus a nonexponential integral term may be written as a purely discrete resonant sum by evaluating at long times the integral term by the steepest descents method, and then expanding the resulting expression in terms of resonant states. This leads to a representation that is valid along the exponential and the inverse power in time regimes. A model calculation using the $\ensuremath{\delta}$ potential allows us to make a comparison of the expansion with numerical integrations in terms of continuum wave functions and, in the long time regime, with an exact analytic expression of the integral term obtained using the steepest descents method. The results demonstrate that resonant states give a correct description of the long-time behavior of decay.

Physical Review AJul 3, 200745 citations

Comment on "Anomalous Spreading of Power-Law Quantum Wave Packets"

R. S. Mendes, Celia Anteneodo

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 $t^{-β}$ law (with $β<1$ ), slower than the usual $t^{-1}$ one, including the particular case $(ln t)^2/t$ not correctly discussed in F. Lillo and R. N. Mantegna (Phys. Rev. Lett. 84, 1061 (2000)).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 $t^{-β}$ law (with $β<1$), slower than the usual $t^{-1}$ one, including the particular case $(ln t)^2/t$ not correctly discussed in F. Lillo and R. N. Mantegna (Phys. Rev. Lett. 84, 1061 (2000)).

arXiv:cond-mat/0003366Mar 22, 20000 citations

Comment on “Resonant Spectra and the Time Evolution of the Survival and Nonescape Probabilities”

W. van Dijk, Yukihisa Nogami

Physical Review LettersJan 17, 200313 citations

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

Manabu Miyamoto

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.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.

arXiv:quant-ph/0207067Physical Review AAug 7, 200311 citations

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)

and nonescape

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)

and

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 $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}1}$.

t^{-β}

law (with

β<1

), slower than the usual

t^{-1}

one, including the particular case

(ln t)^2/t

not correctly discussed in F. Lillo and R. N. Mantegna (Phys. Rev. Lett. 84, 1061 (2000)).

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

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.