Suppose that the speed of an electron traveling $2.0 \mathrm {~km} / \mathrm { s }$ is known to an accuracy of 1 part in $10 ^ { 5 }$ (i.e., within $0.0010 \%$ ). What is the least possible uncertainty within which we can determine the position of this electron? ( $\left. m _ { \text {electron } } = 9.11 \times 10 ^ { - 31 } \mathrm {~kg} , h = 6.626 \times 10 ^ { - 34 } \mathrm {~J} \cdot \mathrm { s } \right)$
A) $2.0 \mathrm {~mm}$
B) $14 \mathrm {~cm}$
C) $5.8 \mathrm {~mm}$
D) $14 \mathrm {~mm}$
E) $8.5 \mathrm {~nm}$

Question

Quizplus · Accepted Answer

The uncertainty in the electron's position can be estimated using the Heisenberg uncertainty principle, which states $\Delta x \Delta p \geq \frac{h}{4\pi}$, where $\Delta x$ is the uncertainty in position, $\Delta p$ is the uncertainty in momentum, and $h$ is Planck's constant. Given the speed and its accuracy, we can calculate $\Delta p$ and then find $\Delta x$.

Suppose That the Speed of an Electron Traveling 2.0 km/s2.0 \mathrm {~km} / \mathrm { s }2.0 km/s

Suppose That the Speed of an Electron Traveling $2.0 \mathrm {~km} / \mathrm { s }$