Final answer:
The minimum uncertainty in the energy of a relatively long-lived excited state of an atom can be calculated using the Heisenberg Uncertainty Principle. In this case, the minimum uncertainty in the energy is approximately 2.21 × 10^(-31) J.
Step-by-step explanation:
The minimum uncertainty in the energy of a relatively long-lived excited state of an atom can be calculated using the Heisenberg Uncertainty Principle. According to this principle, the minimum uncertainty in the energy (ΔE) is related to the minimum uncertainty in time (Δt) as ΔE × Δt ≥ h, where h is the Planck's constant (approximately 6.63 × 10^(-34) J s).
In this case, the lifetime of the excited state (Δt) is given as 3.00 ms (or 3.00 × 10^(-3) s). Thus, the minimum uncertainty in the energy (ΔE) can be calculated as ΔE ≥ (h / Δt) = (6.63 × 10^(-34) J s) / (3.00 × 10^(-3) s) = 2.21 × 10^(-31) J.
Therefore, the minimum uncertainty in the energy of the atom's relatively long-lived excited state is approximately 2.21 × 10^(-31) J.