Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling

We solve a form of the Bloch equation for the density matrix in a manner that identifies a ''quantum'' correction to the classical potential. This correction is given by the difference between the Bohm self-potential and a form of the Wigner potential that have both been used in previous simulations of semiconductor devices. The net potential appearing in the density is then a nonlocally smoothed average of the total semiclassical potential, which itself is composed of the classical potential and the quantum correction. However, the effective potential, which appears in hydrodynamic equations for device modeling, is shown to be the difference between this nonlocally smoothed potential and the local value of the potential. The various definitions that have appeared in the literature for the quantum potential and the effective potential are compared and it is demonstrated how a connection exists among these various definitions.