American Journal of Physics

An exhaustive analysis of the asymptotic time dependence of wave packets in one dimension

K. Unnikrishnan

Published Jul 1, 1998• 15 citations

Abstract

It is shown that the amplitude of a square-integrable wave packet in one dimension must decrease with time asymptotically as either t−1/2+μ, 0⩽μ<1/4, or as t−1/2 ln t. The different patterns reflect the different ways in which the wave function at t=0 falls off at large |x|; μ=0 corresponds to a decrease faster than |x|−1, the logarithmic form to a decrease exactly as |x|−1, and μ>0 to a decrease as |x|−1+2μ.

DOI: 10.1119/1.18918

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American Journal of Physics

An exhaustive analysis of the asymptotic time dependence of wave packets in one dimension

K. Unnikrishnan

Published Jul 1, 1998• 15 citations

Publisher

Abstract

DOI: 10.1119/1.18918

Related papers

Evolution of a wave packet scattered by a one-dimensional potential

А. Ж. Хачатрян, Al.G. Alexanyan, V. A. Khoetsyan +1 more

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.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.

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math-phCross-listedarXiv:2512.21423Dec 26, 20250 citations

A one-dimensional scattering problem off a

\\delta

-shaped potential is solved analytically and the time development of a wave packet is derived from the time-dependent Schrödinger equation. The exact and explicit expression of the scattered wave packet supplies us with interesting information about the "time delay" by potential scattering in the asymptotic region. It is demonstrated that a wave packet scattered by a spin-flipping potential can give us quite a different value for the delay times from that obtained without spin-degrees of freedom.

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.

k

. We prove that at large times, the wave function is approximately equal to the superposition of two wave packets traveling in opposite directions, which results in trajectories with approximately constant asymptotic momentum

k

and asymptotic energy

\pm c^2\sqrt{m^2+k^2}

, with

m

the rest mass of the particle and

c

the speed of light. The sign of the asymptotic energy is determined by the initial position of the particle. Particles with negative energy will have an asymptotic velocity that is in the opposite direction of their momentum. The proof uses the stationary phase approximation method, for which we establish a rigorous error bound.