The minimum uncertainty in the speed of an electron in an one dimensional region of length $2 a_{O}$ (Where $a_{0}=$ Bohr radius 52.9 pm) is ______ $\mathrm{km} \mathrm{s}^{-1}$. (Given: Mass of electron $=9.1 \times 10^{-31} \mathrm{~kg}$, Planck's constant $\mathrm{h}=6.63 \times 10^{-34} \mathrm{Js}$)

JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Atomic Structure

The minimum uncertainty in the speed of an electron in an one dimensional region of length

2 a_{O}

(Where

a_{0} =

Bohr radius 52.9 pm) is ______

km s^{- 1}

.
(Given: Mass of electron

= 9.1 \times 1 0^{- 31} kg

, Planck's constant

h = 6.63 \times 1 0^{- 34} Js

)

Answer: 548

Using Heisenberg's uncertainty principle:

Δ x \times Δ p \geq \frac{h}{4 π}

For minimum uncertainty (equality holds):

Δ x \times m Δ v = \frac{h}{4 π}

Solving for

Δ v

Δ v = \frac{h}{4 π \times Δ x \times m} = \frac{h}{4 π \times 2 a _{0} \times m}

Substituting values:

Δ v = \frac{6.63 \times 1 0 ^{- 34}}{4 \times 3.14 \times 2 \times 52.9 \times 1 0 ^{- 12} \times 9.1 \times 1 0 ^{- 31}}

= 548273 m/s = 548.3 km/s \approx 548 km/s

JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Integer answer