The nonequilibrium density matrix for a test particle in a harmonic lattice under the influence of an external force is obtained exactly. This is done by propagating the equilibrium density matrix (Boltzmann statistics) of the whole lattice, through the lattice propagator, involving the external interaction on the test particle, and finally constructing a single-particle reduced density matrix for the test particle. The general result is then specialized to the case of the Schrödinger chain-oscillator model with one of its particles charged (test particle) while the rest are taken to be neutral. Under conditions of weak coupling, it is shown that the system of test particles can be treated as an independent thermodynamic system. The resulting test-particle nonequilibrium density matrix is essentially a drifted equilibrium density matrix. In this limit the nonequilibrium density matrix for the brownian quantum oscillator under the influence of an external force is obtained. Average values for the nonequilibrium energy, momentum and displacement in the case of the oscillator and the free particle are obtained. The quantum effects only enter the average oscillator energy and this through the equilibrium expression for the energy. The rest of the expressions are identical with the corresponding classical results.