A new numerical technique for following the evolution of collisionless matter under the influence of gravity is proposed. Matter is modeled as a Schrodinger field obeying the coupled Schrodinger and Poisson equations. The de Broglie wavelength, lambda(deB), associated with this field enters as a free parameter and is tuned according to the specifications of the simulation one is doing. In the limit lambda(deB) --> 0 (a limit which would require infinite computing power) the equations reduce to the coupled Vlasov and Poisson equations as they should. Our method can handle multiple streams in phase space and is competitive in terms of computation time with particle-mesh N-body simulations. Results from a simple one-dimensional collapse of a self-gravitating object as well as a two-dimensional simulation of a cold dark matter universe are used to illustrate the viability of the technique.