The x component of the velocity of an electron (m = 9.11 × 10^-³¹ kg)is known to be between 100 m/s and 300 m/s.Which one of the following is a true statement concerning the uncertainty in the x coordinate of the electron? A)The maximum uncertainty is about 10⁶ m. B)The minimum uncertainty is about 3 × 10^-⁷ m. C)The maximum uncertainty is about 6 × 10^-⁷ m. D)The minimum uncertainty is about 3 × 10^-⁹ m. E)The maximum uncertainty is about 6 × 10^-⁹ m.

Question

Quizplus · Accepted Answer

The uncertainty principle, formulated by Werner Heisenberg, can be used to calculate the minimum uncertainty in position ($\Delta x$) given the uncertainty in momentum ($\Delta p$). The principle is given by $\Delta x \Delta p \geq \frac{\hbar}{2}$, where $\hbar$ is the reduced Planck's constant ($\approx 1.0545718 \times 10^{-34} m^2 kg / s$). The uncertainty in the momentum can be calculated using the range of velocities given (100 m/s to 300 m/s) and the mass of the electron ($9.11 \times 10^{-31} kg$). The uncertainty in velocity ($\Delta v$) is $300 m/s - 100 m/s = 200 m/s$, so $\Delta p = m \Delta v = 9.11 \times 10^{-31} kg \times 200 m/s = 1.822 \times 10^{-28} kg \cdot m/s$. Using the uncertainty principle:$$\Delta x \geq \frac{\hbar}{2\Delta p} \approx \frac{1.0545718 \times 10^{-34} m^2 kg / s}{2 \times 1.822 \times 10^{-28} kg \cdot m/s} \approx \frac{1.0545718 \times 10^{-34}}{3.644 \times 10^{-28}} m \approx 2.89 \times 10^{-7} m$$This calculation shows that the minimum uncertainty in the x coordinate of the electron is about $3 \times 10^{-7} m$ or $3 \times 10^{-7}$ meters, which corresponds to choice B.

The X Component of the Velocity of an Electron (M