Question

If uncertainty in position and momentum are equal, then uncertainty in velocity is

• A. $\frac{1}{2m}\sqrt{\frac{h}{\pi}}$
• B. $\sqrt{\frac{h}{2\pi}}$
• C. $\frac{1}{m}\sqrt{\frac{h}{\pi}}$
• D. $\sqrt{\frac{h}{\pi}}$

Answer 1

The Correct Option is A

The problem requires finding the uncertainty in velocity when the uncertainties in position and momentum are equal. This involves using Heisenberg's Uncertainty Principle, which states that the product of the uncertainties in position (\( \Delta x \)) and momentum (\( \Delta p \)) is always greater than or equal to \( \frac{h}{4\pi} \), where \( h \) is Planck's constant:

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

From the given condition that the uncertainties in position and momentum are equal:

\(\Delta x = \Delta p\)

Substituting this into the inequality:

\((\Delta x)^2 \geq \frac{h}{4\pi}\)

Solve for \(\Delta x\):

\(\Delta x \geq \sqrt{\frac{h}{4\pi}}\)

Now, since \(\Delta x = \Delta p\) , we have:

\(\Delta p = \sqrt{\frac{h}{4\pi}}\)

The momentum (\( p \)) is related to velocity (\( v \)) by the relation:

\(p = m \cdot v\)

For uncertainties, this translates to:

\(\Delta p = m \cdot \Delta v\)

Substitute for \(\Delta p\) to find the uncertainty in velocity (\(\Delta v\)):

\(m \cdot \Delta v = \sqrt{\frac{h}{4\pi}}\)

\(\Delta v = \frac{1}{m}\sqrt{\frac{h}{4\pi}}\)

Upon simplifying the right side, note that:

\(\Delta v = \frac{1}{2m}\sqrt{\frac{h}{\pi}}\)

Therefore, the uncertainty in velocity when uncertainties in position and momentum are equal is:

\(\frac{1}{2m}\sqrt{\frac{h}{\pi}}\)

Hence, the correct answer is: \(\frac{1}{2m}\sqrt{\frac{h}{\pi}}\).