On the Interchangeability of Spin Matrix and Orbital Angular Momentum Operators in the Dirac Theory of the Electron

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 matrix and orbital angular momentum operators. Fractional orbital angular momentum is interesting since it partially attributes electron spin to the flow of momentum in space around the spin axis. To develop this idea further, the simpler problem of an electron confined in a 3-dimensional harmonic oscillator is formulated here as a Klein-Gordon equation expressed in terms of raising and lowering operators. Two alternate Dirac equations are then obtained for the oscillator depending on whether they contain a raising or lowering operator. It is shown solutions to both these equations describe an electron having the same energy and spin but differ in that one solution contains fractional orbital angular momentum and the other does not. It is also shown that the fraction of orbital angular momentum present in spin depends on the velocity of the oscillator indicating the further interchangeability of the spin and orbital angular momentum operators through Lorentz transformations.