Question

The uncertainty in momentum of an electron is $1\times 10^{-5} kg m/s.$ The uncertainty in its position will be (Given, $h = 6.62 \times 10^{-34} kg m^2/s)$

• A. $1.05 \times 10^{-28} m$
• B. $1.05 \times 10^{-26} m$
• C. $5.27 \times 10^{-30} m$
• D. $5.25 \times 10^{-28} m$

Answer 1

The Correct Option is C

To find the uncertainty in the position of an electron given the uncertainty in its momentum, we can use the Heisenberg Uncertainty Principle. This principle is defined as:

\Delta x \cdot \Delta p \geq \frac{h}{4\pi}

where:

• \Delta x is the uncertainty in position,
• \Delta p is the uncertainty in momentum,
• h is the Planck's constant.

Let's insert the given values:

• \Delta p = 1 \times 10^{-5} \, \text{kg m/s}
• h = 6.62 \times 10^{-34} \, \text{kg m}^2/\text{s}

Rearranging the formula to find \Delta x:

\Delta x \geq \frac{h}{4\pi \Delta p}

Substitute the values:

\Delta x \geq \frac{6.62 \times 10^{-34}}{4\pi \times 1 \times 10^{-5}}

Calculating the above expression:

\Delta x \geq \frac{6.62 \times 10^{-34}}{12.5664 \times 10^{-5}}

\Delta x \geq \frac{6.62}{12.5664} \times 10^{-29}

\Delta x \geq 0.5265 \times 10^{-29}

\Delta x \geq 5.27 \times 10^{-30} \, \text{m}

Therefore, the uncertainty in the position of the electron, according to the Heisenberg Uncertainty Principle, is closest to 5.27 \times 10^{-30} \, \text{m}. Thus, the correct answer is 5.27 \times 10^{-30} \, \text{m}.