The concept of thermal ratchets is extended to the system governed by quantum mechanics. We study a tight-binding model with an asymmetric periodic potential contacting with a heat bath under an external oscillating field as a specific example of quantum ratchet. The dynamics of a density operator of this system are studied numerically using the quantum Liouville equation. The finite net current is found in the nonequilibrium steady state, which originates from purely quantum mechanical effects. The direction of the current varies with parameters, in contrast with the classical thermal ratchets.