ZnO quantum wires are grown on patterned silicon substrates using plasma-assisted molecular beam epitaxy. A crystallographic etch process exposes the Si(111) planes from Si(001) substrates so that V-grooves are formed. The large misfit between the Si(111) and ZnO(0001) lattice constants leads to low ZnO film quality after growth. Subsequent thermal annealing improves the electronic and structural quality of the film substantially. In the center region of the wire we find a strong photoluminescence intensity enhancement, which arises from carrier localization in the ZnO quantum wire. Two-dimensional self-consistent calculations of the electronic structure show excellent agreement with the experimental results.