# Spreading of gaussian wave packets as a result of the uncertainty principle

I am having a hard time making this excercise

A Gaussian wave packet is associated with an electron localized at time $$t = 0$$ to within a distance of $$10^{−10} m$$ . Show that this wave packet will have spread to twice its size after a time $$t ∼ 10^{−16} s$$.

I tried using the uncertainty principle to get an expression for $$\Delta p$$ depending of $$\Delta t$$ so i took the derivative $$d(E(p)) = \frac{p}{m} dp$$. With this i get an expression for $$\Delta p$$:

$$\Delta p \geq \frac{hm}{2\pi p\Delta t}$$

and so also for $$\Delta x$$

$$\Delta x \geq \frac{p \Delta t}{m}$$

But i don't think i can use this expression to show what is asked since I don't know what the value of $$p$$ is.

Can someone help me?

From Heisenberg's uncertainty principle you get the uncertainty of momentum ($$\Delta p$$). Then, from $$p=mv$$ you can get the uncertainty of velocity ($$\Delta v$$). Finally, from this you can get the additional uncertainty of position after a time $$10^{-16}\text{s}$$. The result should be $$10^{-10}\text{m}$$.