Free Schrodinger Wigner transport ($w_t+\xi \cdot {\nabla }_{x}w=0$)

For $i{\Psi }_t=-\Delta \Psi /2$, differentiate the two-point product $\Psi (x+y/2)\overline{\Psi }(x-y/2)$. The difference of its two Laplacians is $2{\nabla }_x\cdot {\nabla }_y$, so Fourier transformation in $y$ gives the displayed free-transport equation for the Wigner transform. Its solution is $w(x,\xi ,t)=w(x-t\xi ,\xi ,0)$.