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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?

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1 Answer 1

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May be you are thinking too complicated.

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 $10^{-10}\text{m}$.

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