We observe the quantum coherent dynamics of atomic spinor wave packets in the double-well potentials of a far-off-resonance optical lattice. With appropriate initial conditions the system Rabi oscillates between the left and right localized states of the ground doublet, and at certain times the wave packet corresponds to a coherent superposition of these mesoscopically distinct quantum states. The atom/optical double-well potential is a flexible and powerful system for further study of quantum coherence, quantum control, and the quantum/classical transition.