gen-define-the-uncertainty-principle-and-give-one-short-worked-e-6a64

Define the uncertainty principle and give one short worked example.

# Heisenberg's Uncertainty Principle The **Heisenberg Uncertainty Principle** states that certain pairs of physical properties cannot be simultaneously known with arbitrary precision. The most common form relates position and momentum: $$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$$ where: - Δx = uncertainty in position - Δp = uncertainty in momentum - ℏ = h/2π ≈ 1.055 × 10⁻³⁴ J·s (reduced Planck's constant) ## Worked Example **Problem:** An electron's position is measured with an uncertainty of Δx = 1.0 × 10⁻¹⁰ m (roughly the size of an atom). What is the minimum uncertainty in its momentum? **Solution:** Using the uncertainty principle: $$\Delta p \geq \frac{\hbar}{2\Delta x}$$ $$\Delta p \geq \frac{1.055 \times 10^{-34}\text{ J·s}}{2(1.0 \times 10^{-10}\text{ m})}$$ $$\Delta p \geq 5.3 \times 10^{-25} \text{ kg·m/s}$$ **Interpretation:** This momentum uncertainty corresponds to a velocity uncertainty of about 580,000 m/s for an electron (using Δv = Δp/m). This shows why atomic electrons cannot have well-defined orbits—confining them to atomic dimensions creates huge uncertainties in their momentum!