2) This problem continues with the analysis of the Gaussian wave packet.
a) Calculate the probability density $|\Psi(x,t)|^2$ and express it in terms
of $\omega = \sqrt{a/(1+\Omega^2t^2)}$, where $\Omega = 2a\hbar/m$.
b) Sketch or plot $|\Psi(x,0)|^2$ as a function of $x$. Select a time $t > 0$ and sketch again as a
function of $x$. Discuss what is happening.
c) Find $<x>$, $<x^2>$ and $\sigma_x$.
d) Find $<p>$, assume $<p^2> = a\hbar^2$ and calculate $\sigma_p$.
e) Check the uncertainty principle, and find at what time the system comes closest to the
uncertainty limit.
