Fractional Angular Momenta in Electron Beams and Hydrogen-Like Atoms

TL;DR

This paper extends the concept of fractional angular momenta from relativistic electron beams to hydrogen-like atoms, revealing new effects like fractional Gouy phase and wave-particle duality.

Contribution

It applies a method for calculating fractional angular momenta in electron beams to solutions of the Dirac equation for hydrogen-like atoms, uncovering novel angular momentum effects.

Findings

Fractional angular momenta appear in hydrogen-like atoms.

Fractional Gouy phase is observed in electron beams.

Fractional wave-particle duality is identified in atoms and beams.

Abstract

In an earlier letter [Ducharme \textit{et al.} Phys. Rev. Lett. \textbf{126}, 134803 (2021)], a solution to the Dirac equation for a relativistic Gaussian electron beam showed that for a diverging beam the spin of each electron is the sum of fractional contributions from both the spin angular momentum (SAM) and orbital angular momentum (OAM) operators. Fractional angular momenta emerge when eigenstates of the Dirac equation can be decomposed into two terms of opposite spin. Each of these terms being eigenstates of both the SAM and OAM operators. Building on this understanding, the same method used to calculate fractional angular momenta in beams is applied here to solutions of the Dirac equation for hydrogen-like atoms. The results strengthen the idea that factorization of the Klein-Gordon equation using Dirac matrices equation does more than introduce spin, it also produces a specific mixing of the angular momentum states leading to the presence of fractional angular momenta and related effects such as fractional Gouy phase in the case of beams and fractional wave-particle duality in both atoms and beams.