Question

The minimum uncertainty in the speed of an electron in an one dimensional region of length $2 a _0$
(Where $a _0=$ Bohr radius $52.9 \,pm$ ) is _____$km \,s ^{-1}$.
(Given : Mass of electron $=9.1 \times 10^{-31} kg$,
Planck's constant $h =6.63 \times 10^{-34} \,Js$ )

Answer 1

Heisenberg's uncertainty principle
$ \Delta x \times \Delta p _{ x } \geq \frac{ h }{4 \pi} $
$ \Rightarrow 2 a _0 \times m \Delta v _{ x }=\frac{ h }{4 \pi}(\text { minimum })$
$ \Rightarrow \Delta v _{ x }=\frac{ h }{4 \pi} \times \frac{1}{2 a _0} \times \frac{1}{ m } $
$ =\frac{6.63 \times 10^{-34}}{4 \times 3.14 \times 2 \times 52.9 \times 10^{-12} \times 9.1 \times 10^{-31}}$
$ =548273\, ms ^{-1} $
$=548.273 \,km \,s ^{-1} $
$=548 \,km \,s ^{-1}$