Question

If the unrtainty in velocity of an electron $(\Delta v)$ is $0.1 \text{ m/s}$, the unrtainty in its position $(\Delta x)$ is (given: $m_e = 9.1 \times 10^{-31} \text{ kg}$)

• A. $2.02 \times 10^{-4} \text{ m}$
• B. $4.04 \times 10^{-4} \text{ m}$
• C. $5.79 \times 10^{-4} \text{ m}$
• D. $8.42 \times 10^{-4} \text{ m}$

Hint:

Heisenberg’s Principle: There is a fundamental limit to how precisely we can know both position and momentum: $\Delta x \cdot \Delta p \geq \frach4\pi$.

Answer 1

The Correct Option is C

Heisenberg's uncertainty principle is: \[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \] Since momentum $p = mv$, we substitute $\Delta p = m \Delta v$: \[ \Delta x = \frac{h}{4\pi m \Delta v} \] Given:

• $h = 6.6 \times 10^{-34} \text{ J s}$
• $m = 9.1 \times 10^{-31} \text{ kg}$
• $\Delta v = 0.1 \text{ m/s}$

Now plug into the formula: \[ \Delta x = \frac{6.6 \times 10^{-34}}{4 \cdot 3.1416 \cdot 9.1 \times 10^{-31} \cdot 0.1} = \frac{6.6 \times 10^{-34}}{1.1426 \times 10^{-30}} \approx 5.78 \times 10^{-4} \text{ m} \] So the correct option is (3).

Answer 2

Step 1: Understand the uncertainty principle
Heisenberg's uncertainty principle states:
\(\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\)
where \(\Delta x\) is uncertainty in position,
\(\Delta p = m \Delta v\) is uncertainty in momentum,
and \(h = 6.626 \times 10^{-34} \, \text{Js}\) is Planck’s constant.

Step 2: Calculate uncertainty in momentum \(\Delta p\)
Given:
\(\Delta v = 0.1 \, \text{m/s}\),
\(m_e = 9.1 \times 10^{-31} \, \text{kg}\)
\(\Delta p = m_e \times \Delta v = 9.1 \times 10^{-31} \times 0.1 = 9.1 \times 10^{-32} \, \text{kg·m/s}\)

Step 3: Calculate uncertainty in position \(\Delta x\)
Using:
\(\Delta x \geq \frac{h}{4 \pi \Delta p}\)
Substitute values:
\(\Delta x \geq \frac{6.626 \times 10^{-34}}{4 \pi \times 9.1 \times 10^{-32}}\)
Calculate denominator:
\(4 \pi \times 9.1 \times 10^{-32} \approx 1.142 \times 10^{-30}\)
Then:
\(\Delta x \geq \frac{6.626 \times 10^{-34}}{1.142 \times 10^{-30}} = 5.8 \times 10^{-4} \, \text{m}\)

Step 4: Conclusion
The uncertainty in the position of the electron is approximately \(5.79 \times 10^{-4} \, \text{m}\).