Why are Gaussian wave packets the only ones saturating the uncertainty principle?

[ralqs]

Is there an intuitive way to see why Gaussian wave packets are the only ones that satisfy $\Delta x \Delta p = \frac{\hbar}{2}$?

 

[Meir Achuz]

There is no 'intuitive way', but it can be shown mathematically that the Gaussian gives the minimal [tex[\Delta x \Delta p[/tex].