Wave packet (10-30 min)

We consider the wave function (wave packet)

$$\Psi(x)= \frac{1}{\sqrt{\sqrt{\pi a^2}}} \exp{\left(-\frac{x^2}{2a^2}\right)}.$$ (5)

1. Show that this wave function is normalized (remember what normalization means!)

2. Using this wave function, calculate the expectation values $\langle x^2\rangle$, $\quad \langle p^2\rangle$, and their product $\langle x^2\rangle \cdot \langle p^2\rangle$. You have to use the integral $\int_{-\infty}^{\infty}dy y^2e^{-a^2y^2}=\sqrt{\pi}/(2a^3)$.