The correct option is **A. ∆x ∆Px ≥ h**. ### Detailed Explanation The uncertainty principle is a fundamental concept in quantum mechanics, formulated by Werner Heisenberg. It states that there are inherent limitations in our ability to simultaneously know certain pairs of physical properties of a particle, such as position and momentum. 1. **Understanding the Variables**: - **∆x**: This represents the uncertainty in the measurement of the position of a particle along the x-axis. It quantifies how precisely we can know the particle's position. - **∆Px**: This represents the uncertainty in the measurement of the linear momentum of the particle along the x-axis. Linear momentum (P) is defined as the product of mass (m) and velocity (v), so P = mv. The uncertainty in momentum reflects how precisely we can know the particle's momentum. 2. **The Uncertainty Principle**: The uncertainty principle can be mathematically expressed as: \[ ∆x ∆Px ≥ \frac{h}{4\pi} \] where \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \, \text{Js} \). For simplicity, in many contexts, the relation is often stated as: \[ ∆x ∆Px ≥ h \] This means that the product of the uncertainties in position and momentum cannot be smaller than a certain value, which is a fundamental limit imposed by the nature of quantum systems. 3. **Why Option A is Correct**: - The inequality \( ∆x ∆Px ≥ h \) indicates that as we try to measure the position of a particle more precisely (reducing ∆x), the uncertainty in its momentum (∆Px) must increase, and vice versa. This is a reflection of the wave-particle duality of matter, where particles exhibit both wave-like and particle-like properties. - The principle highlights the intrinsic limitations of measurement at the quantum level, emphasizing that we cannot have perfect knowledge of both position and momentum simultaneously. ### Why the Other Options are Incorrect: - **Option B: ∆x ∆Px = 0**: - This option suggests that it is possible to have no uncertainty in both position and momentum, which contradicts the uncertainty principle. If both uncertainties were zero, it would imply that we have perfect knowledge of both properties, which is impossible according to quantum mechanics. - **Option C: ∆x ∆Px < h**: - This option implies that the product of the uncertainties can be less than Planck's constant, which is also incorrect. The uncertainty principle states that the product must be greater than or equal to a certain value, not less. - **Option D: ∆x ∆Px = ∞**: - This option suggests that the product of the uncertainties is infinite, which is not a valid interpretation of the uncertainty principle. While the uncertainties can be large, they are not infinite, and the principle provides a specific lower bound. ### Common Pitfalls: - Confusing the inequality with an equality. The uncertainty principle is an inequality, and it is crucial to remember that the product of uncertainties cannot be less than a certain value. - Misunderstanding the implications of the principle. It is not just about measurement errors; it reflects a fundamental property of quantum systems. ### Revision Summary: - The uncertainty principle states that ∆x ∆Px ≥ h, indicating a fundamental limit on the precision of simultaneous measurements of position and momentum. - As one uncertainty decreases, the other must increase, reflecting the wave-particle duality of matter. - Options suggesting zero uncertainty or that the product can be less than h are incorrect. - Remember that the uncertainty principle is a key concept in quantum mechanics, emphasizing the limits of measurement at the quantum level.