Physics Letters A 377 (2012) 33–38 Contents lists available at SciVerse ScienceDirect Physics Letters A www.elsevier.com/locate/pla An “Airy gun”: Self-accelerating solutions of the time-dependent Schrödinger equation in vacuum ✩ Alex Mahalov, Sergei K. Suslov ∗ School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA article info abstract Article history: Received 11 September 2012 Received in revised form 24 October 2012 Accepted 25 October 2012 Available online 5 November 2012 Communicated by V.M. Agranovich Keywords: Schrödinger equation Nonspreading wave packets Diffraction-free optical beams Airy functions Second Painlevé transcendents Wigner function We consider generalizations of the Berry and Balazs nonspreading and accelerating solution of the time-dependent Schrödinger equation in empty space, which has been experimentally demonstrated in paraxial optics. In particular, we show that the original nonspreading wave packet is unstable. An explicit variation of the initial Airy-state evolves into the self-accelerating and self-compressing solution presented here. Quasi-diffraction-free finite energy Airy beams that are more realistic for experimental study are obtained by analytic continuation and their Wigner function is evaluated. Nonlinear generalizations related to second Painlevé transcendents are briefly discussed. Published by Elsevier B.V. Although Ehrenfest’s theorem of quantum mechanics (embody- ing Newton’s second law for classical particles) suggests that no wave packet can accelerate in free space, Berry and Balazs [11] introduced nonspreading Airy beams, which do accelerate with- out any external force, somewhat contrary to conventional wisdom (see [14,23,28,56] for further exploration of different aspects of this result). These nonspreading and freely accelerating wave pack- ets have been demonstrated in both one- and two-dimensional configurations as quasi-diffraction-free optical beams [49,50] thus generating a considerable interest to this phenomenon (see [1,3– 5,7,15,18,20,24,32,33,38,44,45] and the references therein). This is why, we would like to elaborate on this matter from a theoreti- cal perspective paying particular attention to generalizations and instability of Airy beams, which also deserve an experimental con- firmation (see also [21] and the references therein). 1. Accelerating nonspreading wave packets The time-dependent Schrödinger equation for a free particle (or the normalized paraxial wave equation in optics [49] also known as the parabolic equation [25]), i ψ t + ψ xx = 0, (1.1) ✩ This research was partially supported by an AFOSR grant FA9550-11-1-0220. * Corresponding author. E-mail addresses: mahalov@asu.edu (A. Mahalov), sks@asu.edu (S.K. Suslov). URL: http://hahn.la.asu.edu/~suslov/index.html (S.K. Suslov). by the following substitution ψ(x, t ) = e ig(x−2gt 2 /3)t g 1/3 F ( g 1/3 ( x − gt 2 )) , g = a/2 (1.2) (a is the acceleration) can be reduced to the Airy equation: F ′′ = zF , z = g 1/3 ( x − gt 2 ) , (1.3) whose bounded solutions are the Airy functions F = k Ai(z) (up to a multiplication constant k) with well-known asymptotics as z → ±∞ [25,43]. Combining with the familiar Galilean transformation (2.3), ψ(x, t ) = e i (x−vt /2)v/2 χ (x − vt , t ) (1.4) ( v is the velocity), one obtains a more general solution of this type ψ(x, t ) = e i (x−vt /2)v/2+ig(x−vt −2gt 2 /3)t × g 1/3 F ( g 1/3 ( x − vt − gt 2 )) . (1.5) These freely accelerating Airy beams were found by Berry and Bal- azs [11] in the context of quantum mechanics (see also [14,23,28, 56] for further discussions of their properties). Explicit solutions (1.2) and (1.5) in terms of the Airy func- tion reveal certain remarkable features. The quantity |ψ(x, t )| 2 not only remains unchanged in form but also constantly accelerates in empty space. There is no violation of Ehrenfest’s theorem because the Airy function is not square integrable and cannot represent the probability density for a single particle. The center of mass of this solution does not exist. What accelerates in the Airy packet is not 0375-9601/$ – see front matter Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physleta.2012.10.041