Physics Letters A 310 (2003) 101–109 www.elsevier.com/locate/pla Shrinking quantum packets in one dimension V.V. Dodonov ∗,1 , M.A. Andreata Departamento de Física, Universidade Federal de São Carlos, Via Washington Luiz km 235, 13565-905 São Carlos, SP, Brazil Received 14 December 2002; accepted 31 January 2003 Communicated by V.M. Agranovich Abstract We give examples of “shrinking” free wave packets with zero initial velocity in one dimension. Considering different measures of spatial extension of wave packets, we show that degree of shrinking of “two-hump” packets corresponding to initial even coherent states is greater than for their mixed counterparts and for odd pure states.  2003 Elsevier Science B.V. All rights reserved. PACS: 03.65.-w; 03.75.-b Keywords: Quantum packets; Even/odd coherent states; Pure and mixed states; Probability flux; Measures of spatial extension 1. Introduction Although propagation of quantum packets in free space was discussed for decades in numerous papers, beginning with [1], and in almost all textbooks on quantum mechanics, this subject is not exhausted yet. For recent few years, several new interesting effects in propagation and scattering (reflection) of packets of matter waves have been discovered [2–7]. Here we are studying peculiar features of the dynamics of one-dimensional packets, whose probability density initially has more than one maximum. Our interest to this problem is motivated by the recent article [7]. * Corresponding author. E-mail addresses: vdodonov@df.ufscar.br (V.V. Dodonov), pmauro@df.ufscar.br (M.A. Andreata). 1 On leave from Lebedev Physical Institute and Moscow Institute of Physics and Technology, Russia. It is well known that, in the absense of external fields, initially localized packets spread unlimitedly with the course of time (for the most recent discussions see, e.g., [8]; the problem of controlling the shape of packets with the aid of external fields was considered, e.g., in [9]). However, it was shown recently [7] that certain “ring-shaped” packets in two dimensions experience initially some small “contraction”, and only after certain finite interval of time they begin to spread monotonously. More precisely, considering a free evolution in d space dimensions of the packet whose initial form depends only on radius as (we use dimensionless variables, assuming, in particular, m = ¯ h = 1) (1) ψ d (r ; 0) = N d r 2 exp ( -r 2 ) , the authors of [7] showed that the two-dimensional case is distinguished in the sense that the mean value of radius decreases with time in some interval 0 < 0375-9601/03/$ – see front matter  2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00288-3