 Solution

Set $G(x,y,t)=\Psi (x+y/2,t)\overline{\Psi }(x-y/2,t)$. The Schrödinger equation and its conjugate givesince ${\nabla }_x={\nabla }_{+}+{\nabla }_{-}$ and ${\nabla }_y=({\nabla }_{+}-{\nabla }_{-})/2$. In the Wigner transform, integration by parts turns ${\nabla }_y$ into $i\xi$, yielding the free transport equationAlong its characteristics $x(t)=x_0+t\xi$, the value is constant. ThusFor rapidly decreasing smooth data every integration by parts is justified; weaker data follow distributionally.