Relativistic quantum-mechanical description of twisted paraxial electron and photon beams

Abstract

The analysis of twisted (vortex) paraxial photons and electrons is fulfilled in the framework of relativistic quantum mechanics. The use of the Foldy-Wouthuysen representation radically simplifies the description of relativistic electrons and clarifies fundamental properties of twisted particles. It is demonstrated that the twisted and other structured photons are luminal. Their subluminality apparently takes place because the photon energy is also contributed by a hidden motion. This motion is vanished by averaging and disappears in the semiclassical description based on expectation values of the momentum and velocity operators. It is proven that semiclassical quanta of structured light are subluminal and massive. The quantum-mechanical and semiclassical descriptions of twisted and other structured electrons lead to similar results. The new effect of a quantization of the velocity and the effective mass of the structured photon and electron is predicted. This effect is observable for the photon. The twisted and untwisted semiclassical photon and electron modeled by the centroids are considered in the accelerated and rotating noninertial frame. The coincidence of their inertial masses with kinematic ones is shown. The orbital magnetic moment of the Laguerre-Gauss electron does not depend on the radial quantum number.

General quantum-mechanical solution for twisted electrons in a uniform magnetic field

“…A correspondence between the relativistic quantum-mechanical equations in the FW representation and the paraxial wave equations has been established in Refs. [28,31]. The correspondence is very similar for photons and electrons.…”

“…For electrons, it can be obtained from the Dirac equation in the Foldy-Wouthuysen (FW) representation provided that |p ⊥ | p [28]. The paraxial wave function of free electrons and photons characterizes the Laguerre-Gauss (LG) beams and reads [26,29,30]…”

“…A transformation of Hamiltonian equations in the FW representation to the paraxial form has been considered in Refs. [28,31,45]. Squaring Eq.…”

Paraxial wave function and Gouy phase for a relativistic electron in a uniform magnetic field

“…This property increases the similarity between the wave eigenfunctions of the relativistic Landau electron in the uniform magnetic field and the twisted electron in a free space. In the latter case, the eigenfunction reads [9][10][11]…”

Twisted particles in heavy-ion collisions

“…Photons with OAMs more than 10000 [1] and electrons with OAMs up to 1000 [2] have been obtained. Twisted photons have nonzero effective masses [3]. Charged twisted particles can be recognized by their dynamics [4,5], magnetic moments [6], and specific effects in external fields [7][8][9].…”