Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0003366

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

R. S. Mendes, Celia Anteneodo

Published Mar 22, 2000• 0 citations
arXivHTMLPDF

Abstract

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)).

DOI
10.48550/arxiv.cond-mat/0003366

Related papers

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.

arXiv:quant-ph/0207067•Physical Review A•Aug 7, 2003•11 citations

On the asymptotic decay law for wave packets in free space

K. Unnikrishnan

The Cauchy wave packet is introduced, whose probability density in free space depends on time asymptotically as [ln(t)]6/t3, thus displaying a remarkable departure from the commonly expected t−3 behavior.

American Journal of Physics•Jun 1, 1997•16 citations

Comments on ‘‘Illustration of resonances and the law of exponential decay in a simple quantum mechanical problem,’’ by Herbert Massmann [Am. J. Phys. 5 3, 679–683 (1985)]

D. S. Onley, Anita Kumar

The time dependence of the wave packet spreading is investigated for the case of a particle confined in a one−dimensional box. (AIP)

American Journal of Physics•Jun 1, 1991•6 citations

Critical wave-packet dynamics in the power-law bond disordered Anderson model

R. P. A. Lima, F. A. B. F. de Moura, M. L. Lyra +1 more

We investigate the wave-packet dynamics of the power-law bond disordered one-dimensional Anderson model with hopping amplitudes decreasing as ${H}_{nm}\ensuremath{\propto}{\ensuremath{\mid}n\ensuremath{-}m\ensuremath{\mid}}^{\ensuremath{-}\ensuremath{\alpha}}$. We consider the critical case $(\ensuremath{\alpha}=1)$. Using an exact diagonalization scheme on finite chains, we compute the participation moments of all stationary energy eigenstates as well as the spreading of an initially localized wave packet. The eigenstates multifractality is characterized by the set of fractal dimensions of the participation moments. The wave packet shows a diffusivelike spread developing a power-law tail and achieves a stationary nonuniform profile after reflecting at the chain boundaries. As a consequence, the time-dependent participation moments exhibit two distinct scaling regimes. We formulate a finite-size scaling hypothesis for the participation moments relating their scaling exponents to the ones governing the return probability and wave-function power-law decays.

arXiv:cond-mat/0502353•Physical Review B•Jun 24, 2005•13 citations

Anomalous Spreading of Power-Law Quantum Wave Packets

Fabrizio Lillo, Rosario N. Mantegna

We introduce power-law tail quantum wave packets. We show that they can be seen as eigenfunctions of a Hamiltonian with a physical potential. We prove that the free evolution of these packets presents an asymptotic decay of the maximum of the wave packets which is anomalous for an interval of the characterizing power-law exponent. We also prove that the number of finite moments of the wave packets is a conserved quantity during the evolution of the wave packet in the free space.

arXiv:cond-mat/9912387•Physical Review Letters•Feb 7, 2000•22 citations

Spreading Wave Packets—A Cautionary Note

Barry R. Holstein, Arthur R. Swift

The spreading of a quantum mechanical wave packet is studied from two apparently equivalent points of view. For certain classes of wave functions, one prescription predicts spreading while the other does not. Reasons for this anomaly are discussed.

American Journal of Physics•Jun 1, 1972•20 citations
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)).
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.
We investigate the wave-packet dynamics of the power-law bond disordered one-dimensional Anderson model with hopping amplitudes decreasing as ${H}_{nm}\ensuremath{\propto}{\ensuremath{\mid}n\ensuremath{-}m\ensuremath{\mid}}^{\ensuremath{-}\ensuremath{\alpha}}$. We consider the critical case $(\ensuremath{\alpha}=1)$. Using an exact diagonalization scheme on finite chains, we compute the participation moments of all stationary energy eigenstates as well as the spreading of an initially localized wave packet. The eigenstates multifractality is characterized by the set of fractal dimensions of the participation moments. The wave packet shows a diffusivelike spread developing a power-law tail and achieves a stationary nonuniform profile after reflecting at the chain boundaries. As a consequence, the time-dependent participation moments exhibit two distinct scaling regimes. We formulate a finite-size scaling hypothesis for the participation moments relating their scaling exponents to the ones governing the return probability and wave-function power-law decays.