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arXiv:quant-ph/0404057 · Physical Review A

Initial wave packets and the various power-law decreases of scattered wave packets at long times

Manabu Miyamoto

Published Apr 5, 2004• 10 citations
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Abstract

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 $\ensuremath{\psi}(x,t)$.

DOI
10.1103/physreva.69.042704

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

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The long-time behavior of scattered wave packets \ensuremathψ(x,t)\ensuremath{\psi}(x,t)\ensuremathψ(x,t) from a finite-range potential is investigated, by assuming \ensuremathψ(x,t)\ensuremath{\psi}(x,t)\ensuremathψ(x,t) to be initially located outside the potential. It is then shown that \ensuremathψ(x,t)\ensuremath{\psi}(x,t)\ensuremathψ(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 \ensuremathψ(x,t)\ensuremath{\psi}(x,t)\ensuremathψ(x,t) exhibits the asymptotic behavior t\ensuremath−3∕2{t}^{\ensuremath{-}3∕2}t\ensuremath−3∕2, while another behavior like t\ensuremath−5∕2{t}^{\ensuremath{-}5∕2}t\ensuremath−5∕2 can also appear for another \ensuremathψ(x,t)\ensuremath{\psi}(x,t)\ensuremathψ(x,t).
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 time behavior of a wave packet, transmitted through a potential barrier, is investigated for all times assuming it to be initially located outside the asymmetric barrier. The survival probability of the initial state, which characterizes the decay of the unstable quantum system in three time domains, is discussed. The presence of initial, exponential, and long-time regimes is a general feature of the decay process. We analytically found the three crucial points of time that separate these regimes. These characteristic times are set by simple formulae, including momentum distribution. It is then shown that the radius of convergence of the cumulant expansion for the survival probability is always small. A long-time behavior of survival probability is described by the asymptotic series whose coefficients decrease with increasing particle energy but at different rates. To be specific, we have examined the exactly solvable model of tunneling through a finite range triangular barrier.
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.
We consider the evolution of a wave packet that is made up of a group of the wave functions describing the stationary scattering process and tunnels through a one-dimensional potential of arbitrary form. As the main characteristics of the time difference of the tunnelling process, use is made of the propagation speed of the wave-packet maximum. We show that the known Hartman formula for the tunnelling time corresponds to the wave packet with a wavenumber-uniform spectral composition in the case, when the phase and transmission coefficient modulus dispersions are taken into account only in the linear approximation. The amplitude of the main peak of the transmitted wave intensity is proven to be independent of the tunnelling time and is determined by the transmission coefficient of the spectral component at the carrier frequency and the spectral width of the wave packet. In the limit of an infinitely wide potential barrier the amplitude of the wave-packet maximum is shown to tend to zero slower than the tunnelling time tends to its asymptotic value, i.e., indeed we deal with the paradox of an infinitely large propagation speed of a wave disturbance through the barrier.
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)).