Journal of the Physical Society of Japan

Characterization of the Power Decays of Survival Probabilities at Long Times for Free Quantum Particle System

Manabu Miyamoto

Published Jan 1, 2003• 0 citations

Abstract

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.

DOI: 10.1143/jpsjs.72sc.69

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Journal of the Physical Society of Japan

Characterization of the Power Decays of Survival Probabilities at Long Times for Free Quantum Particle System

Manabu Miyamoto

Published Jan 1, 2003• 0 citations

Publisher

Abstract

DOI: 10.1143/jpsjs.72sc.69

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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.

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)).

We consider a hypothetical experiment where a particle is released at time

t=0

in a fixed initial state. A detector makes instantaneous quantum measurements at regular intervals of time

\tau

, and keeps doing so till it detects the particle, say on the

n^{\rm th}

observation, at time

t=n\tau

. We take

\tau

to be small but do not take the limit of continuous measurements

\tau\to 0

. We examine the survival probability

P(t)

that the particle is un-detected till time

t

. Here we present results for a simple model of a free particle in a one-dimensional box and find that interesting features such as power-law tails are obtained for the survival probability. We provide a perturbative calculation which compares very well with numerical results. Some exact results are obtained for a mean-field model. In the limit of continuous observations, as expected, one always gets a Zeno-like effect where the particle is never detected.