Gaussian and Airy wave packets of massive particles with orbital angular momentum

Abstract

While wave-packet solutions for relativistic wave equations are oftentimes thought to be approximate (paraxial), we demonstrate that there is a family of such solutions, which are exact, by employing a null-plane (light-cone) variables formalism. A scalar Gaussian wave-packet in transverse plane is generalized so that it acquires a well-defined z-component of the orbital angular momentum (OAM), while may not acquire a typical "doughnut" spatial profile. Such quantum states and beams, in contrast to the Bessel ones, may have an azimuthal-angle-dependent probability density and finite quantum uncertainty of the OAM, which is determined by the packet's width. We construct a well-normalized Airy wave-packet, which can be interpreted as a one-particle state for relativistic massive boson, show that its center moves along the same quasi-classical straight path and, what is more important, spreads with time and distance exactly as a Gaussian wave-packet does, in accordance with the uncertainty principle. It is explained that this fact does not contradict the well-known "non-spreading" feature of the Airy beams. While the effective OAM for such states is zero, its quantum uncertainty (or the beam's OAM bandwidth) is found to be finite, and it depends on the packet's parameters. A link between exact solutions for the Klein-Gordon equation in the null-plane-variables formalism and the approximate ones in the usual approach is indicated, generalizations of these states for a boson in external field of a plane electromagnetic wave are also presented.