Unified dynamics of electrons and photons via the quantum relativistic phenomena of Zitterbewegung and spin-orbit interaction

Cody Leary 

Example manifestation of the spin-orbit interaction between an electron propagating in a cylindrically symmetric waveguide generated by an electric field, or a photon propagating in a waveguide made of a transparent material with a variable index of refraction. For either particle, “orbital” clover patterns in the particle’s wavefunction form inside the waveguide, which rotate under the spin-orbit interaction, with the rotation direction controlled by the particle’s spin.

When an electron propagates in a cylindrically symmetric waveguide, it experiences two physical effects that require both relativity and quantum mechanics to explain. The first, known as Zitterbewegung (German for “trembling motion”), is related to the fact that the position of an electron cannot fundamentally be determined more precisely than the scale of its own Compton wavelength λc = h / m c (otherwise, the energy required to measure the electron’s position that precisely would give rise to spontaneous electron-positron pair creation, and one would not tell which electron was the “original” one in the resulting chaos). The second effect is known as the spin-orbit interaction, which occurs whenever a particle with spin propagated through a spatially varying electric field. Since special relativity predicts that an electric field is partially transformed to a magnetic field form the frame of reference of the electron moving through it, the spin magnetic moment of the electron interacts with this magnetic field, which influences the electron’s orbit (see above figure for an example of this). Surprisingly, it has been recently predicted that a photon propagating in a waveguide of similar symmetry will interact with the waveguide medium in an identical manner as do the electrons described above. This implies that a geometrical mechanism encompassing the struc- ture of both relativity or quantum mechanics may be responsible for this “coincidence”. Recent Wooster graduates have made this analogy more concrete by helping to carrying out calculations predicting how the energy, phase velocity, and spatial wavefunctions of electrons and photons change as a result of these subtle yet exotic interactions. Most recently, we have begun to ex- tend this analogy to spherical geometries, comparing the well-known physics of a Dirac electron bound in an atomic potential with that of a photon trapped via total internal reflection in a micron-scale glass sphere, in order to see if the Zitterbewegung and spin-orbit analogies persist. In this project, we will continue this work via both pencil/paper calculations and compu- tational/numerical simulations in Mathematica. In another possible direction, we will create a simpler quantum mechanical “toy” model that incorporates the physics of interest while avoiding the obfuscating complications present in the complete relativistic quantum theory.