Worked example: An electron is localised to Δx = 0.1 nm (about an atom's size). What is the minimum uncertainty in its momentum and velocity? Step 1: Δp ≥ ℏ/(2Δx) = 1.055×10⁻³⁴ / (2 × 10⁻¹⁰) = 5.28×10⁻²⁵ kg·m/s. Step 2: Δv = Δp/m = 5.28×10⁻²⁵ / 9.11×10⁻³¹ ≈ 5.8×10⁵ m/s. Step 3: Compare with a thermal electron (~10⁵ m/s): the uncertainty alone exceeds typical speeds — confining an electron to an atom forces it to move relativistically fast, which is why atomic electrons have keV-scale kinetic energies. Step 4: Energy–time check: an excited state with lifetime 10⁻⁸ s has ΔE ≥ ℏ/(2×10⁻⁸) ≈ 3.3×10⁻⁸ eV natural width. Conclusion: squeezing position pumps up momentum — the uncertainty principle sets the energy scale of atoms.