An electron inside a hydrogen atom is confined to within a space of about $0.12 \mathrm {~nm}$. What is the uncertainty in this electron's speed? (melectron $= 9.11 \times 10 ^ { - 31 } \mathrm {~kg} , h = 6.626 \times 10 ^ { - 34 } \mathrm {~J} \cdot \mathrm { s }$ )
A) $4.8 \times 106 \mathrm {~m} / \mathrm { s }$
B) $3.1 \times 10 ^ { 5 } \mathrm {~m} / \mathrm { s }$
C) $3.1 \mathrm {~km} / \mathrm { s }$
D) $9.6 \times 10 ^ { 5 } \mathrm {~m} / \mathrm { s }$
E) $4.8 \mathrm {~m} / \mathrm { s }$

Question

Quizplus · Accepted Answer

The uncertainty in position $\Delta x$ is $0.12 \, \text{nm} = 0.12 \times 10^{-9} \, \text{m}$. Using the Heisenberg uncertainty principle $\Delta x \Delta p \geq \frac{h}{4\pi}$, where $\Delta p = m \Delta v$, we find $\Delta v \approx \frac{h}{4\pi m \Delta x}$. Substituting the given values, $\Delta v \approx 9.6 \times 10^5 \, \text{m/s}$.

An Electron Inside a Hydrogen Atom Is Confined to Within 0.12 nm0.12 \mathrm {~nm}0.12 nm

An Electron Inside a Hydrogen Atom Is Confined to Within $0.12 \mathrm {~nm}$