The correct option for the uncertainty principle relation is **C. ∆x ∆Px < h**. ### Explanation of the Correct Answer The uncertainty principle, formulated by Werner Heisenberg, is a fundamental concept in quantum mechanics that describes the limitations of simultaneously knowing certain pairs of physical properties of a particle, such as position and momentum. The principle states that the more precisely we know a particle's position (∆x), the less precisely we can know its momentum (∆Px), and vice versa. The mathematical expression of the uncertainty principle is given by: \[ \Delta x \Delta P_x \geq \frac{h}{4\pi} \] Where: - ∆x is the uncertainty in position, - ∆Px is the uncertainty in momentum, - h is Planck's constant (approximately \(6.626 \times 10^{-34} \, \text{Js}\)). This means that the product of the uncertainties in position and momentum is always greater than or equal to a constant value, which is a small number (specifically, \(\frac{h}{4\pi}\)). ### Why Option C is Correct - **C. ∆x ∆Px < h**: This option is correct because it reflects the idea that the product of the uncertainties can be less than Planck's constant, but it cannot be zero or infinite. The inequality indicates that there is a limit to how precisely we can know both properties at the same time. ### Why the Other Options are Incorrect - **A. ∆x ∆Px ≥ h**: This option is incorrect because it suggests that the product of the uncertainties is greater than or equal to Planck's constant, which is not the case. The correct formulation includes a factor of \(\frac{h}{4\pi}\), which is much smaller than \(h\). - **B. ∆x ∆Px = 0**: This option is incorrect because it implies that we can know both the position and momentum of a particle with perfect accuracy, which contradicts the uncertainty principle. If either uncertainty were zero, it would imply infinite uncertainty in the other property, which is not physically possible. - **D. ∆x ∆Px = ∞**: This option is also incorrect. While it is true that if we have no information about a particle's position or momentum, the uncertainties could be very large, the uncertainty principle does not state that the product of uncertainties is infinite. Instead, it establishes a specific relationship that limits how much we can know about both properties simultaneously. ### Summary of Key Points - The uncertainty principle states that the product of the uncertainties in position and momentum is limited by a constant value. - The correct relationship is \(\Delta x \Delta P_x \geq \frac{h}{4\pi}\), indicating that the uncertainties cannot be simultaneously minimized to zero. - Understanding the uncertainty principle is crucial for grasping the behavior of particles at the quantum level, where classical mechanics fails. - The principle highlights the fundamental limits of measurement in quantum mechanics, emphasizing the wave-particle duality of matter. This understanding is essential for students preparing for exams in physics, particularly in topics related to quantum mechanics and the behavior of particles at microscopic scales.