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arXiv:0801.0188 · American Journal of Physics

The evolution of free wave packets

Mark Andrews

Published Nov 18, 2008• 26 citations
arXivHTMLPDF

Abstract

We discuss four general features of the force-free evolution of wave packets: (1) The spatial spread of a packet changes with time in a simple way. (2) For sufficiently short durations (related to the spread in the momentum of the packet) the probability distribution will move with uniform speed and little change in shape. (3) After a sufficiently long time (related to the initial spatial spread) the wave function converges to a simple form that is simply related to the momentum distribution of the packet. In this asymptotic regime the shape of the probability distribution no longer changes, and its scale increases linearly with the time. (4) There is an infinite denumerable set of simple wave packets (the Hermite-Gauss packets) that do not change shape at any time during their evolution.

DOI
10.1119/1.2982628

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We discuss four general features of the force-free evolution of wave packets: (1) The spatial spread of a packet changes with time in a simple way. (2) For sufficiently short durations (related to the spread in the momentum of the packet) the probability distribution will move with uniform speed and little change in shape. (3) After a sufficiently long time (related to the initial spatial spread) the wave function converges to a simple form that is simply related to the momentum distribution of the packet. In this asymptotic regime the shape of the probability distribution no longer changes, and its scale increases linearly with the time. (4) There is an infinite denumerable set of simple wave packets (the Hermite-Gauss packets) that do not change shape at any time during their evolution.
The usual textbook discussions of the velocity and spreading of a wave packet are based on special cases and approximations which have a limited range of validity. In this paper more general relations are obtained by considering the time dependence of the moments of a wave packet. Both a packet of real sinusoidal plane waves (a real pulse, e.g., sound) and a packet of complex exponential plane waves (e.g., the quantum-mechanical wave function of a free particle) are considered. The results are similar if the real wave packet contains no standing wave components. It is shown that the centroid of a wave packet moves at a constant velocity which is a weighted average of the group velocity. The spreading of the wave packet depends on the product of the variance of the group velocity and the square of the time interval before or after the time of minimum packet width. The packet moves and spreads like a collection of particles in free flight, in which the velocities of the particles are the group velocities of the wave components of the packet.
Abstract The features of the spatial propagation and the evolution of a free correlated wave packet describing the process of particle motion are considered. It is shown that, unlike an “ordinary” noncorrelated packet, for which the process of spatial “spreading” and defocusing begins with the instant of its creation, in the case of a moving correlated packet, in the initial state, a self-simulated self-compression process occurs to the collapse state at a given distance; then the accelerated “spreading” mode is implemented. Some methods for forming such a packet and the possibility of using it in applied problems for particle and energy transport and for the aims of medical hyperthermia are considered.
A general relationship is established between wave-packet evolution and quantization in classically integrable systems. Because of wave-packet spreading, one cannot simply take the Fourier transform of the time evolution of a wave packet. Instead, one must propagate the wave packet using the actions as Hamiltonians. The energy eigenvalues which result are the Einstein-Brillouin-Keller values, and new forms for the eigenfunctions appear. These are free of caustic singularities, and represent averages of wave packets over the invariant torus.