# Uncertainty principle and an electron

question is:
use the uncertainty principle to show that if an electron were confined inside and atomic nucleus, diameter of $$2 x 10^{-15}m$$, it would have to be moving relativistically i.e. more than 0.1c.
what i have done is the following:
$$\Delta x \Delta p = \frac{\hbar}{2}$$
then i set $$\Delta x = 2 x 10^{-15}$$ and solved for $$\Delta p$$

from this i dont know how to solve for the speed, i considered replacing $$\Delta p$$ with $$\gamma m\Delta v$$ but this doesnt seem to work with the gamma in there.
what is a better way to solve this?
thanks

## Answers and Replies

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robphy
Homework Helper
Gold Member
Must you actually solve for the speed? [it can be done, since you can write $$\gamma$$ in terms of v... but do you need to?]
While v>0.1c may characterize "relativistic", is there another way?

this was the hint that was given, that relativistic speeds were > 0.1c, but if you could give me a hint on comparing something else it would be appreciated.

robphy
Homework Helper
Gold Member
What is the corresponding inequality for $$\gamma$$?

i dont know of anything similar, but gamma will be less than or equal to 1. is this close to the right track?

Find $\frac{\gamma v}{c}$ (= $\frac{\Delta p}{mc}$)

Squre both sides and solve for v/c.

gamma will be less than or equal to 1.
v < c
Therefore, gamma > 1

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