Consider the Schrödinger equation

$$i \partial_{t} \Psi=-\frac{1}{2} \Delta \Psi, \quad x \in \mathbb{R}^{n}, t>0$$

for complex-valued solutions $\Psi(x, t)$ and where $\Delta$ is the Laplacian.

(a) Derive, by using a Fourier transform and its inversion, the fundamental solution of the Schrödinger equation. Obtain the solution of the initial value problem

$$\begin{aligned} i \partial_{t} \Psi=-\frac{1}{2} \Delta \Psi, & x \in \mathbb{R}^{n}, \quad t>0 \\ \Psi(x, 0)=f(x), & x \in \mathbb{R}^{n} \end{aligned}$$

as a convolution.

(b) Consider the Wigner-transform of the solution of the Schrödinger equation

$$w(x, \xi, t)=\frac{1}{(2 \pi)^{n}} \int_{\mathbb{R}^{n}} \Psi\left(x+\frac{1}{2} y, t\right) \bar{\Psi}\left(x-\frac{1}{2} y, t\right) e^{-i y \cdot \xi} \mathrm{d}^{n} y$$

defined for $x \in \mathbb{R}^{n}, \xi \in \mathbb{R}^{n}, t>0$. Derive an evolution equation for $w$ by using the Schrödinger equation. Write down the solution of this evolution equation for given initial data $w(x, \xi, 0)=g(x, \xi)$.