Given below are two statements: Statement (I): It is impossible to specify simultaneously with arbitrary precision, 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 \( \geq \sqrt{\frac{h}{\pi}} \times \frac{1}{2m} \).
In the light of the above statements, choose the correct answer from the options given below:
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The Heisenberg Uncertainty Principle is fundamental in quantum mechanics, stating that we cannot simultaneously know the exact position and momentum of a particle. The uncertainty in velocity is derived from the uncertainties in position and momentum.
- Statement I: This is a restatement of the Heisenberg Uncertainty Principle, which asserts that it is impossible to precisely determine both the position and the momentum of a particle simultaneously. This statement is true. - Statement II: The Heisenberg Uncertainty Principle provides the relationship between the uncertainty in position (\( \Delta x \)) and momentum (\( \Delta p \)), given by: \[ \Delta x \Delta p \geq \frac{h}{4\pi} \] For an electron, if the uncertainties in position and momentum are equal, the uncertainty in velocity \( \Delta v \) can be expressed as: \[ \Delta v = \frac{\Delta p}{m} \geq \sqrt{\frac{h}{\pi}} \times \frac{1}{2m} \] This statement is also correct. Therefore, the correct answer is \( \boxed{(2)} \) Both Statement I and Statement II are true.
