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 λ_{c} = 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.