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.
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