## Relation of radius/diameter size to precision

$\Delta p \Delta x\geq \frac{h}{4\pi }$

Fenared Ortiz 3K
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Joined: Wed Nov 16, 2016 3:03 am

### Relation of radius/diameter size to precision

A question asks :
An electron is confined to the size of a magnesium atom(radius=150pm). What is min. uncertainty in its velocity? Another electron is confined to the size of a sodium atom (radius>150pm). In which atom can the velocity of the electron be known with the highest precision and why?

For the first part I used the Heisenberg Indeterminacy equation and got 1.93x10^5 m/s.
What im confused about is how can i prove which radius gives the higher precision and why.

Ryan Sydney Beyer 2B
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Joined: Fri Sep 29, 2017 7:07 am
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### Re: Relation of radius/diameter size to precision

The Heisenberg Uncertainty Equation can be stated as following::

$\Delta p \Delta x \geq \frac{h}{4\pi }$

We can either know momentum to extreme precision or position to extreme precision but either way it has to be greater than or equal to Planck's Constant divided by 4pi. So in the problem, as the uncertainty in position rises, the uncertainty in velocity would decrease.

This can be shown by rearranging the above equation to show that as position represented by delta x increases, then uncertainty in velocity would decrease::

$\Delta v = \frac{h}{4\pi \Delta xm}$

For the problem then, the atom with a radius of 150pm would have greater uncertainty in velocity than the atom with a radius greater than 150pm. The electron confined to the sodium atom is the atom in which the velocity would be known with the highest precision.

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### Re: Relation of radius/diameter size to precision

Should we convert all the units to meters or picometers before doing this problem?

Mikaila 3E
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Joined: Fri Sep 29, 2017 7:07 am

### Re: Relation of radius/diameter size to precision

Converting units is a preference thing. I prefer to convert my units at the beginning so I don't forget at the end but if you prefer to work only in certain units, then that is fine too. The important part is to have the correct final answer in the correct units.