Rotating Gaussian wave packets in weak external potentials

by | Mar 1, 2017 | Seminar Talks

Speaker: Arseni Goussev (Department of Mathematics, Physics, and Electrical Engineering, Northumbria University)

Title: Rotating Gaussian wave packets in weak external potentials

Abstract: Among many motivations to study the time evolution of quantum matter-wave packets two are particularly noteworthy. First, localized wave packets provide the most natural tool for investigating the correspondence between quantum and classical motion. Indeed, while the center of a propagating wave packet traces a trajectory, a concept essential in classical mechanics, its finite spatial extent makes quantum interference effects possible. Second, wave packets may be used as basis functions. That is, any initial state of a quantum system can be represented as a superposition of a number, finite or infinite, of localized wave packets. Thus, an analytical understanding of how each individual wave packet moves through space offers a way to quantitatively describe the time-evolution of an arbitrary, often complex, initial state. Despite a large body of literature on quantum wave packet dynamics, the subject is by no means exhausted.

In a recent paper [1], Dodonov has addressed the time evolution of nonrelativistic two-dimensional Gaussian wave packets with a finite value of mean angular momentum (MAM). The value is the sum of the “external” MAM, related to the center of mass motion, and the “internal” MAM, resulting from the rotation of the wave packet around its center of mass. Among many interesting features of such wave packets is the effect of initial shrinking of packets with strong enough coordinate-momentum correlation.

In my talk, motivated by Ref. [1], I will consider the propagation of a localized two- or three-dimensional rotating Gaussian wave packet in the presence of a weak external potential. The particular focus will be on the time evolution of the internal MAM of the moving wave packet. I will derive, using a semiclassical approximation of the eikonal type, an explicit formula that gives the value of the internal MAM as a function of the propagation time, parameters of the initial wave packet and the external potential. An example physical scenario, in which a two-dimensional particle traverses a tilted ridge barrier, will be considered in full detail. In particular, it will be shown how an initially uncorrelated, rotation-free wave packet may, upon a collision with the potential barrier, acquire a finite internal MAM, and how the magnitude and direction of the MAM are determined by the aspect ratio and orientation of the incident wave packet.

[1] V. V. Dodonov. “Rotating quantum Gaussian packets,” J. Phys. A: Math. Theor. 48, 435303 (2015).