Q1(k). Energy uncertainty of an excited state

Answer in brief

ΔE·τ ≥ ℏ/2: the ground state lives for ever (τ → ∞, ΔE → 0); an excited state decays in about 10⁻⁸ s, so its energy is uncertain by ΔE ≥ ℏ/2τ.

Key points

• Energy–time uncertainty: ΔE·Δt ≥ ℏ/2; for a state, Δt is its lifetime τ.
• Ground state: stable, does not decay, τ → ∞ ⇒ ΔE → 0; its energy is sharp.
• Photoexcited state: decays by emission or radiationless steps, τ ≈ 10⁻⁹ to 10⁻⁸ s (singlet); for τ = 10 ns, ΔE ≥ 5.27 × 10⁻²⁷ J.
• Seen as lifetime (natural) broadening: Δν = 1/2πτ = 15.9 MHz for 10 ns, 159 MHz for 1 ns, only 159 Hz for a phosphorescent triplet (τ = 1 ms).