  We investigate the dynamics of a particle in a confined periodic system---a
time-dependent oscillator confined by infinitely high and moving walls---and
focus on the evolution of the phase of the wavefunction. It is shown that, for
some specific initial states in this potential, the phase of the wavefunction
throughout the cavity depends on the walls motion. We further elaborate a
thought experiment based on interferences devised to detect this form of
single-particle nonlocality from a relative phase. We point out that, within
the non-relativistic formalism based on the Schr\"odinger equation (SE),
detecting this form of nonlocality can give rise to signaling. We believe this
effect is an artifact, but the standard relativistic corrections to the SE do
not appear to fix it. Specific illustrations are given, with analytical results
in the adiabatic approximation, and numerical computations to show that
contributions from high-energy states (corresponding to superluminal
velocities) are negligible.
