### 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, known as complementary variables or canonically conjugate variables, can be known simultaneously. The most commonly discussed pair of these variables is 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 the mass (m) of the particle 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 emphasizes 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 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: "The position of a particle can be known exactly if its momentum is also known exactly."** - This statement contradicts the Heisenberg Uncertainty Principle. If both position and momentum were known exactly, it would imply that both \( \Delta x \) and \( \Delta p \) are zero, which violates the principle. Therefore, this option is incorrect. - **Option C: "The position and momentum of a particle can both be measured with unlimited precision simultaneously."** - This option directly contradicts the Heisenberg Uncertainty Principle. The principle asserts that there is a fundamental limit to the precision of simultaneous measurements of position and momentum. Thus, this option is also incorrect. - **Option D: "The uncertainty in position and momentum is independent of each other and can be measured accurately."** - This statement is misleading. The uncertainty in position and momentum is not independent; they are intrinsically linked by the Heisenberg Uncertainty Principle. If one is measured with high precision, the other must be measured with lower precision. Therefore, this option is incorrect as well. ### Summary of Key Points: - The Heisenberg Uncertainty Principle states that the more precisely one property (position) is known, the less precisely the other property (momentum) can be known. - Mathematically, this is expressed as \( \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \). - This principle highlights a fundamental limit in quantum mechanics, not a limitation of measurement tools. - Understanding this principle is crucial for grasping the behavior of particles at the quantum level. By keeping these points in mind, you can better understand the implications of the Heisenberg Uncertainty Principle in quantum mechanics.