Is there a general relationship between the density operator and phase velocity?

I am trying to solve $int v_pdrho$ for a time-independent, 1-particle density operator $rho$ (acquired, more or less, from a partial trace of complex scalar field operators projected upon the vacuum state, i.e., $psi^{dagger}|0rangle langle0|psi$) and the phase velocity $v_p$ corresponding to an arbitrary element of the particle’s (matter) wave. As such, I was wondering if there is a general relationship between either a 1-particle density operator or a corresponding 1-particle wave-function and the phase velocity?