where ρ 0 (x 0 ) = ρ(x 0 , t = 0), x = (x 0 , t), and x (x 0 , t) = ∂x(x 0 , t) ∂x 0 | t . Note that the "spatial" derivative of x is taken with respect to the labelling coordinate C, which in this case is just the initial value x 0 . The dimensionless quantity x thus becomes a measure of the relative spacing of nearby trajectories over time, as compared to the initial spacing at t = 0. Inserting the expressions for ρ(x, t) [from Eq. ( )] and for x (x 0 , t) into the above wavefunction-based quantum hydrodynamic equations, all reference to the time-evolved wavefunction are now entirely removed. The resultant trajectory-based dynamical PDE