Given below are two statements : Statement (I) : It is impossible to specify simultaneously with arbitrary precision, both the linear momentum and the position of a particle. Statement (II) : If the uncertainty in the measurement of position and uncertainty in measurement of momentum are equal for an electron, then the uncertainty in the measurement of velocity is $$\ge \sqrt{\frac{h}{\pi}} \times \frac{1}{2m}$$. In the light of the above statements, choose the correct answer from the options given below :

Statement I: "It is impossible to specify simultaneously with arbitrary precision, both the linear momentum and the position of a particle."

This is a direct statement of the Heisenberg Uncertainty Principle, which says:

$$\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$$

where $$\Delta x$$ is the uncertainty in position and $$\Delta p$$ is the uncertainty in momentum. Since the product of the two uncertainties has a non-zero lower bound, we cannot make both arbitrarily small at the same time.

So Statement I is true.

Statement II: We are told that the uncertainty in position equals the uncertainty in momentum, i.e., $$\Delta x = \Delta p$$. We need to check whether $$\Delta v \ge \sqrt{\frac{h}{\pi}} \times \frac{1}{2m}$$.

Starting from the Heisenberg Uncertainty Principle:

$$\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$$

Since $$\Delta x = \Delta p$$, we substitute:

$$(\Delta p)^2 \ge \frac{h}{4\pi}$$

Taking the square root of both sides:

$$\Delta p \ge \sqrt{\frac{h}{4\pi}} = \frac{1}{2}\sqrt{\frac{h}{\pi}}$$

Now, momentum $$p = mv$$, so the uncertainty in momentum is related to the uncertainty in velocity by:

$$\Delta p = m \cdot \Delta v$$

Substituting:

$$m \cdot \Delta v \ge \frac{1}{2}\sqrt{\frac{h}{\pi}}$$

Dividing both sides by $$m$$:

$$\Delta v \ge \frac{1}{2m}\sqrt{\frac{h}{\pi}}$$

This can be rewritten as:

$$\Delta v \ge \sqrt{\frac{h}{\pi}} \times \frac{1}{2m}$$

This is exactly what Statement II claims. So Statement II is also true.

Since both statements are true, the correct answer is Option C.