### Correct Option: B **Explanation of the Heisenberg Uncertainty Principle:** The Heisenberg Uncertainty Principle is a fundamental concept in quantum mechanics that describes a limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables or conjugate variables, can be known simultaneously. The most commonly discussed pair of these variables are position (x) and momentum (p). 1. **Understanding Position and Momentum:** - **Position (x)** refers to the location of a particle in space. - **Momentum (p)** is defined as the product of a particle's mass (m) and its velocity (v), expressed mathematically as \( p = mv \). 2. **The Principle Statement:** - The Heisenberg Uncertainty Principle states that the more accurately we know the position of a particle, the less accurately we can know its momentum, and vice versa. This relationship can be mathematically expressed as: \[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \] where \( \Delta x \) is the uncertainty in position, \( \Delta p \) is the uncertainty in momentum, and \( \hbar \) (h-bar) is the reduced Planck's constant, approximately \( 1.055 \times 10^{-34} \, \text{Js} \). 3. **Why Option B is Correct:** - Option B accurately reflects the essence of the Heisenberg Uncertainty Principle. It highlights the trade-off between the precision of measuring position and momentum. If you try to measure the position of a particle very precisely (small \( \Delta x \)), the uncertainty in its momentum (\( \Delta p \)) must increase, leading to a less precise measurement of momentum. This is a fundamental property of quantum systems and is not due to limitations in measurement technology. ### Why the Other Options are Incorrect: - **Option A: "Both position and momentum can be measured with arbitrary precision simultaneously."** - This option is incorrect because it directly contradicts the Heisenberg Uncertainty Principle. The principle explicitly states that there is a limit to how precisely both can be known at the same time. If one is known with high precision, the other must be known with less precision. - **Option C: "The uncertainty in energy and time can be completely eliminated in quantum systems."** - This option is misleading. While there is a relationship between energy and time uncertainties (expressed as \( \Delta E \cdot \Delta t \geq \frac{\hbar}{2} \)), it does not imply that these uncertainties can be eliminated. Instead, it indicates that there is a similar trade-off between the precision of energy and the duration of time over which that energy is measured. - **Option D: "The principle only applies to macroscopic objects and not to subatomic particles."** - This option is incorrect because the Heisenberg Uncertainty Principle is a fundamental aspect of quantum mechanics that specifically applies to subatomic particles. In fact, it is most significant at the quantum level, where the effects of uncertainty are pronounced. Macroscopic objects, due to their larger mass and size, exhibit negligible uncertainty effects, making them appear deterministic. ### Summary of Key Points: - The Heisenberg Uncertainty Principle states that the more precisely one property (position) is measured, the less precisely the complementary property (momentum) can be known. - This principle is mathematically expressed as \( \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \). - It applies specifically to quantum systems and is not a limitation of measurement technology. - Understanding this principle is crucial for grasping the behavior of particles at the quantum level, where classical physics does not apply.