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Physical Review Letters

In- and Outbound Spreading of a Free-Particle s -Wave

Iwo Białynicki‐Birula, M. A. Cirone, Joachim Dahl, Mikhail Vladimirovich Fedorov, Wolfgang P. Schleich

Published Jul 23, 2002• 43 citations
Publisher

Abstract

We show that a free quantum particle in two dimensions with zero angular momentum (s wave) in the form of a ring-shaped wave packet feels an attraction towards the center of the ring, leading first to a contraction followed by an expansion. An experiment to demonstrate this effect is also outlined.
DOI
10.1103/physrevlett.89.060404

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We study the effect of initial ‘shrinking’ of radial ‘ring-shaped’ wave packets with zero initial velocity distribution in two and more dimensions. Considering time evolution of probability densities described by explicit exact solutions of the free Schrödinger equation in d dimensions, we introduce and compare two different families of quantitative measures of ‘expansion’. The measures of the first type are based on the mean values of arbitrary powers of radius, they characterize the ‘total extension’ of the packet. The measures of the second type quantify the ‘internal’ size of the packet (or an effective width of the ‘ring’). We find that the effect of initial ‘in-spreading’, which is very small in two dimensions and absent in higher dimensions with respect to the mean value of radius, is much more pronounced, if one uses mean values of ⟨ r α ⟩ with α < 1 and especially α < 0 as the measure of packet extension. In this sense, the case of two dimensions is not distinguished, and shrinking packets exist in more than two dimensions, as well. On the other hand, we show that the effect of initial ‘shrinking’ or ‘expansion’ strongly depends on the chosen measure of spatial extension (‘total’ or ‘internal’) of the packet. Moreover, the conclusions concerning the initial evolution of the ‘internal’ packet extensions based on the ‘volume’ probability density |ψ| 2 may be sometimes opposite to the conclusions based on the ‘radial probability density’ r d −1 |ψ| 2 .
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.
We consider the time evolution of a special class of wave packets representing the motion of a free particle along a line. In order to make contact with an earlier work on s-waves in two or higher dimensions we confine ourselves to wave packets with a probability density that is symmetric with respect to the origin. Here we concentrate on the phenomenon that an appropriately prepared wave packet is focusing before it displays its familiar behavior of spreading. We derive the condition on the wave packet for this effect to occur: the probability density at the origin has to be concave. The translation of this requirement into Wigner phase space reads: the average of the kinetic energy along the momentum axis has to be negative. In this way we identify the origin of this focusing effect of the wave packet as quantum interference.