On shrinking and expansion of radial wave packets

Abstract

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 .