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AIP conference proceedings

A study of a class of power-law tail quantum wave packets

Fabrizio Lillo

Published Jan 1, 2000• 0 citations
Publisher

Abstract

We study some properties of a class of quantum wave packets with power-law spatial tails analitically. Almost all packets of this class have the property that all the moments of the position operator are infinite. We prove that the free evolution of these packets presents an asymptotic decay of the wave packet maximum which is anomalous with respect to the behavior observed in the free evolution of customary quantum wave packets. We also study the temporal evolution of the tails and we show that the dominant term of the asymptotic expansion is conserved.

DOI
10.1063/1.1303346

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We study some properties of a class of quantum wave packets with power-law spatial tails analitically. Almost all packets of this class have the property that all the moments of the position operator are infinite. We prove that the free evolution of these packets presents an asymptotic decay of the wave packet maximum which is anomalous with respect to the behavior observed in the free evolution of customary quantum wave packets. We also study the temporal evolution of the tails and we show that the dominant term of the asymptotic expansion is conserved.
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)).
We propose a setup for which a power-law decay is predicted to be observable for generic and realistic conditions. The system we study is very simple: a quantum wave packet initially prepared in a potential well with (i) tails asymptotically decaying like ∼ x −2 and (ii) an eigenvalues spectrum that shows a continuous part attached to the ground or equilibrium state. We analytically derive the asymptotic decay law from the spectral properties for generic, confined initial states. Our findings are supported by realistic numerical simulations for state-of-the-art expansion experiments with cold atoms.
Recently, the stochastic nonlinear wave equation describing the dynamics of an observed quantum system has been solved for a free particle [Phys. Rev. A 45, 1347 (1992)]. In this paper, transient effects that occur during a continuous observation of the particle position are discussed. It is shown that the dispersion of the Gaussian wave packet of the particle always decreases in the beginning of the observation, then takes the form of oscillations which, after a short time, decay rapidly to the asymptotic value.
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.