Answer :
(a) The minimum uncertainty in an electron's velocity is, ![4.5\times 10^{4}m/s](https://tex.z-dn.net/?f=4.5%5Ctimes%2010%5E%7B4%7Dm%2Fs)
(b) The minimum uncertainty in a helium atom's velocity is, ![7.9\times 10^{1}m/s](https://tex.z-dn.net/?f=7.9%5Ctimes%2010%5E%7B1%7Dm%2Fs)
Explanation :
According to the Heisenberg's uncertainty principle,
...........(1)
where,
= uncertainty in position
= uncertainty in momentum
h = Planck's constant
And as we know that the momentum is the product of mass and velocity of an object.
![p=m\times v](https://tex.z-dn.net/?f=p%3Dm%5Ctimes%20v)
or,
.......(2)
Equating 1 and 2, we get:
![\Delta x\times m\times \Delta v=\frac{h}{4\pi}](https://tex.z-dn.net/?f=%5CDelta%20x%5Ctimes%20m%5Ctimes%20%5CDelta%20v%3D%5Cfrac%7Bh%7D%7B4%5Cpi%7D)
![\Delta v=\frac{h}{4\pi \Delta x\times m}](https://tex.z-dn.net/?f=%5CDelta%20v%3D%5Cfrac%7Bh%7D%7B4%5Cpi%20%5CDelta%20x%5Ctimes%20m%7D)
(a) Given:
m = mass of electron = ![9.11\times 10^{-31}kg](https://tex.z-dn.net/?f=9.11%5Ctimes%2010%5E%7B-31%7Dkg)
h = Planck's constant = ![6.626\times 10^{-34}Js](https://tex.z-dn.net/?f=6.626%5Ctimes%2010%5E%7B-34%7DJs)
= ![13\AA=13\times 10^{-10}m](https://tex.z-dn.net/?f=13%5CAA%3D13%5Ctimes%2010%5E%7B-10%7Dm)
conversion used : ![(1\AA=10^{-10}m)](https://tex.z-dn.net/?f=%281%5CAA%3D10%5E%7B-10%7Dm%29)
Now put all the given values in the above formula, we get:
![\Delta v=\frac{6.626\times 10^{-34}Js}{4\times 3.14\times (13\times 10^{-10}m)\times (9.1\times 10^{-31}kg)}](https://tex.z-dn.net/?f=%5CDelta%20v%3D%5Cfrac%7B6.626%5Ctimes%2010%5E%7B-34%7DJs%7D%7B4%5Ctimes%203.14%5Ctimes%20%2813%5Ctimes%2010%5E%7B-10%7Dm%29%5Ctimes%20%289.1%5Ctimes%2010%5E%7B-31%7Dkg%29%7D)
![\Delta v=4.5\times 10^{4}m/s](https://tex.z-dn.net/?f=%5CDelta%20v%3D4.5%5Ctimes%2010%5E%7B4%7Dm%2Fs)
The minimum uncertainty in an electron's velocity is, ![4.5\times 10^{4}m/s](https://tex.z-dn.net/?f=4.5%5Ctimes%2010%5E%7B4%7Dm%2Fs)
(b) Given:
m = mass of helium atom = ![6.646\times 10^{-27}kg](https://tex.z-dn.net/?f=6.646%5Ctimes%2010%5E%7B-27%7Dkg)
h = Planck's constant = ![6.626\times 10^{-34}Js](https://tex.z-dn.net/?f=6.626%5Ctimes%2010%5E%7B-34%7DJs)
= ![1.0\AA=1.0\times 10^{-10}m](https://tex.z-dn.net/?f=1.0%5CAA%3D1.0%5Ctimes%2010%5E%7B-10%7Dm)
conversion used : ![(1\AA=10^{-10}m)](https://tex.z-dn.net/?f=%281%5CAA%3D10%5E%7B-10%7Dm%29)
Now put all the given values in the above formula, we get:
![\Delta v=\frac{6.626\times 10^{-34}Js}{4\times 3.14\times (1.0\times 10^{-10}m)\times (6.646\times 10^{-27}kg)}](https://tex.z-dn.net/?f=%5CDelta%20v%3D%5Cfrac%7B6.626%5Ctimes%2010%5E%7B-34%7DJs%7D%7B4%5Ctimes%203.14%5Ctimes%20%281.0%5Ctimes%2010%5E%7B-10%7Dm%29%5Ctimes%20%286.646%5Ctimes%2010%5E%7B-27%7Dkg%29%7D)
![\Delta v=7.9\times 10^{1}m/s](https://tex.z-dn.net/?f=%5CDelta%20v%3D7.9%5Ctimes%2010%5E%7B1%7Dm%2Fs)
The minimum uncertainty in a helium atom's velocity is, ![7.9\times 10^{1}m/s](https://tex.z-dn.net/?f=7.9%5Ctimes%2010%5E%7B1%7Dm%2Fs)