= Solution
Set $G(x,y,t)=\Psi(x+y/2,t)\overline\Psi(x-y/2,t)$. The <Schrödinger equation> and its conjugate give
$$
\partial_tG=\frac i2(\Delta_+-\Delta_-)G=i\nabla_x\cdot\nabla_yG,
$$
since $\nabla_x=\nabla_++\nabla_-$ and $\nabla_y=(\nabla_+-\nabla_-)/2$. In the <Wigner transform>, integration by parts turns $\nabla_y$ into $i\xi$, yielding the free transport equation
$$
\boxed{\partial_tw+\xi\cdot\nabla_xw=0.}
$$
Along its characteristics $x(t)=x_0+t\xi$, the value is constant. Thus
$$
\boxed{w(x,\xi,t)=g(x-t\xi,\xi).}
$$
For rapidly decreasing smooth data every integration by parts is justified; weaker data follow distributionally.
Back to article page
