The minimum uncertainty in the speed of an electron in a one dimensional region of length $2 a_{0}$ (where $a_{0} =$ Bohar radius $52.9$ pm) is $k m s^{- 1}$ .

(Given : Mass of electron $= 9.1 \times 10^{- 31}$ kg, Planck's constant $h = 6.63 \times 10^{- 34} J s$ )

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Solution

Heisenberg's uncertainty principle

$Δ x \times Δ P_{x} \geq \frac{h}{4 π}$

$\Rightarrow 2 a_{0} \times m Δ v_{x} = \frac{h}{4 π}$ (minimum)

$\Rightarrow Δ V_{x} = \frac{h}{4 π} \times \frac{1}{2 a_{0}} \times \frac{1}{m}$

$= \frac{6.63 \times 10^{- 34}}{4 \times 3.14 \times 2 \times 52.9 \times 10^{- 12} \times 9.1 \times 10^{- 31}}$

$= 548273 m s^{- 1}$

$= 548.273 k m s^{- 1}$

$= 548 k m s^{- 1}$

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