Uncertainty principle using ΔEΔt ≥ ħ: A 440-nm spectral line is produced by a transition from an excited state to the ground state. The natural line width of the spectral line is 0.020 pm. The average time the atom spends in the excited state is closest to which of the following? (ħ = 1.055 × 10^-34 J ∙ s = 6.59 × 10^-16 eV ∙ s) A) 5 × 10^-6 s B) 5 × 10^-7 s C) 5 × 10^-⁸ s D) 5 × 10^-9 s E) 5 × 10^-10 s

Question

Quizplus · Accepted Answer

The uncertainty principle, given as ΔEΔt ≥ ħ/2, relates the uncertainty in energy (ΔE) to the uncertainty in time (Δt). The natural line width (Δλ) can be related to the uncertainty in energy through the relation ΔE = hc/λ^2 * Δλ, where h is Planck's constant and c is the speed of light. Given Δλ = 0.020 pm = 0.020 × 10^-12 m and λ = 440 nm = 440 × 10^-9 m, we can calculate ΔE and then use the uncertainty principle to find Δt. Since ħ is given, we can rearrange the formula to Δt = ħ/(2ΔE) to find the average time the atom spends in the excited state. The calculation will show that the time is closest to the value given in option D, 5 × 10^-9 s.

Uncertainty Principle Using ΔEΔt ≥ ħ: a 440-Nm Spectral Line