Q1(l). Uncertainty in the velocity of an electron

Answer in brief

Δv = h/(4πmΔx) = 1.45 × 10⁶ m s⁻¹, comparable with the speed of the electron itself.

Key points

• Given: Δx = 0.4 Å = 4 × 10⁻¹¹ m; m = 9.109 × 10⁻³¹ kg; h = 6.626 × 10⁻³⁴ J s.
• Heisenberg: Δx · mΔv ≥ h/4π ⇒ Δv = h/(4πmΔx) = 1.45 × 10⁶ m s⁻¹.
• Δv is 0.66 of the Bohr velocity (2.19 × 10⁶ m s⁻¹): an electron held within atomic size has no defined trajectory ⇒ orbitals, not orbits.
• Other forms: with ΔxΔp ≈ h, Δv = 1.82 × 10⁷ m s⁻¹; with ℏ, 2.89 × 10⁶ m s⁻¹.
• Same Δx for a 1 g body: Δv = 1.32 × 10⁻²¹ m s⁻¹; the principle matters only for microscopic particles.