An electron moves with a highly measured speed of $3.3 imes 10^5 \mathrm{~m/s}$ within an accuracy

Question

An electron moves with a highly measured speed of $3.3 	imes 10^5 \mathrm{~m/s}$ within an accuracy of $0.001\%$. The fundamental uncertainty in the position of this electron is approximately ($h = 6.626 	imes 10^{-34} \mathrm{~Js}, m_e = 9.1 	imes 10^{-31} \mathrm{~kg}$)

Professor PC Thomas Classes · Answer

(a) : The absolute uncertainty in velocity is $\Delta v = 3.3 	imes 10^5 	imes \frac{0.001}{100} = 3.3 	imes 10^5 	imes 10^{-5} = 3.3 \mathrm{~m/s}$. According to Heisenberg's Uncertainty Principle, $\Delta x \cdot m \Delta v = \frac{h}{4\pi}$. Isolating position uncertainty: $\Delta x = \frac{h}{4\pi m \Delta v} = \frac{6.626 	imes 10^{-34}}{4 	imes 3.141 	imes 9.1 	imes 10^{-31} 	imes 3.3} = \frac{6.626 	imes 10^{-34}}{37.74 	imes 10^{-31}} \approx 1.756 	imes 10^{-5} \mathrm{~m}$. The closest value is $1.76 	imes 10^{-5} \mathrm{~m}$.