Consider the Schrödinger equation

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

where $V$ is a smooth real-valued function.

Prove that, for smooth solutions, the following equations are valid for all $t>0$ :

(i)

$$\int_{\mathbb{R}^{n}}|\psi(t, x)|^{2} d x=\int_{\mathbb{R}^{n}}\left|\psi_{I}(x)\right|^{2} d x$$

(ii)

$$\begin{aligned} &\int_{\mathbb{R}^{n}} \frac{1}{2}|\nabla \psi(t, x)|^{2} d x+\int_{\mathbb{R}^{n}} V(x)|\psi(t, x)|^{2} d x \\ &=\int_{\mathbb{R}^{n}} \frac{1}{2}\left|\nabla \psi_{I}(x)\right|^{2} d x+\int_{\mathbb{R}^{n}} V(x)\left|\psi_{I}(x)\right|^{2} d x \end{aligned}$$