As the size of modern semiconductor devices goes to sub-nanometers, quantum mechanical phenomena become prominent and must be considered in numerical simulations. In 1989, Ancona and Iafrate derived a macroscopic model, called quantum drift-diffusion (QDD) model, which generalizes the classical DD model by incorporating a quantum correction to the electric potential. We derive an equivalent QDD model by expressing carrier densities with potential functions. The finite element method for the new model is positivity-preserving in the sense that discrete carrier densities are always positive. We propose a modified Newton iterative method to solve the nonlinear discrete problem. Numerical experiments for a FinFet device show that the iterative method is convergent for the source-drain bias voltage up to 15V and the source-gate bias voltage up to 5V.