  An observation of neutron-antineutron oscillations ($ n-\bar{n}$), which
violate both $B$ and $B-L$ conservation, would constitute a scientific
discovery of fundamental importance to physics and cosmology. A stringent upper
bound on its transition rate would make an important contribution to our
understanding of the baryon asymmetry of the universe by eliminating the
post-sphaleron baryogenesis scenario in the light quark sector. We show that
one can design an experiment using slow neutrons that in principle can reach
the required sensitivity of $\tau_{n-\bar{n}}\sim 10^{10}s$ in the oscillation
time, an improvement of $\sim10^4$ in the oscillation probability relative to
the existing limit for free neutrons. This can be achieved by allowing both the
neutron and antineutron components of the developing superposition state to
coherently reflect from mirrors. We present a quantitative analysis of this
scenario and show that, for sufficiently small transverse momenta of
$n/\bar{n}$ and for certain choices of nuclei for the $n/\bar{n}$ guide
material, the relative phase shift of the $n$ and $\bar{n}$ components upon
reflection and the $\bar{n}$ annihilation rate can be small.
