  Bell-type experiments that test correlated observables typically involve
measurements of spin or polarization on multi-particle systems in singlet
states. These observables are all non-commuting and satisfy an uncertainty
relation. Theoretically, the non-commuting nature should be independent of
whether the singlet state consists of multiple particles or a single particle.
Recent experiments in single neutron interferometry have in fact demonstrated
this. In addition, if Bell-type inequalities can be found for experiments
involving spin and polarization, the same should be true for experiments
involving other non-commuting observables such as position and momentum as in
the original EPR paper. As such, an experiment is proposed to measure (quantum
mechanically) position and momentum for a single oscillator as a means for
deriving a Bell-type inequality for these correlated observables. The
experiment, if realizable, would shed light on the basic nature of matter,
perhaps pointing to some form of self-entanglement, and would also help to
further elucidate a possible mechanism behind the Heisenberg uncertainty
principle. Violation of these inequalities would, in fact, offer yet another
confirmation of the principle.
