Question

The uncertainty in the location of circulating electron is equal to its de Broglie wavelength. The minimum percentage error in its measurement of velocity under this circumstance will be approximately:

• A. $4$
• B. $8$
• C. $12$
• D. $16$

Answer 1

The correct option is B $8$

The uncertainty in position is equal to de Broglie wavelength.
$⟹\Delta x=\lambda =\frac{h}{mv}$ ....... (1)

The Heisenberg's uncertainty principle is $\Delta x\Delta v\ge \frac{h}{4\pi m}$ ...... (2)
From equations (1) and (2),

$(\frac{h}{mv}\Delta v\ge \frac{h}{4\pi m})$

Thus, $\frac{\Delta v}{v}\ge \frac{1}{4\pi }$

Hence, minimum percentage error $=\frac{\Delta v}{v}\times 100=\frac{100}{4\pi }=7.96\approx 8$.