Uncertainty principle using ΔEΔt ≥ ħ: A laser produces a beam of 4000-nm light. A shutter allows a pulse of light, 30 ps in duration, to pass. Which of the following is closest to the uncertainty in the energy of a photon in the pulse? (ħ = 1.055 × 10^-34 J ∙ s = 6.59 × 10^-16 eV ∙ s) A) 2 × 10^-6 eV B) 2 × 10^-5 eV C) 2 × 10^-4 eV D) 2 × 10^-3 eV E) 2 × 10^-2 eV

Question

Quizplus · Accepted Answer

The uncertainty principle states that ΔEΔt ≥ ħ. We are given the duration of the pulse (Δt = 30 ps = 30 × 10^-12 s). The energy of a photon can be found using the equation E = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength. We are given the wavelength (λ = 4000 nm = 4 × 10^-7 m). Using these values, we can calculate the energy of a photon as follows: E = hc/λ = (6.626 × 10^-34 J ∙ s)(3.00 × 10⁺⁸ m/s)/(4 × 10^-7 m) E = 4.97 × 10^-19 J/photon To compute the uncertainty in energy, we need to multiply the minimum uncertainty in time by the rate of change of energy with respect to time. Since the light pulse is very short, we can assume that the uncertainty in time is approximately equal to the duration of the pulse (Δt). The rate of change of energy with respect to time can be approximated by dividing the energy by the duration of one cycle of the electromagnetic wave, which is equal to the period (T = λ/c). Thus, we have: ΔE = ħ/(2Δt) × 2π/T = ħπΔf/Δt where Δf is the uncertainty in frequency, which is related to the uncertainty in energy by ΔE = ħΔf. Solving for ΔE, we get: ΔE = ħ/(2Δt) × 2πc/λ = ħπΔc/Δλ where Δc is the uncertainty in the speed of light, which is negligible, and Δλ is the uncertainty in wavelength. Using the wavelength and duration given above, we get: Δλ = cΔt/λ = (3.00 × 10⁺⁸ m/s)(30 × 10^-12 s)/(4 × 10^-7 m) Δλ = 2.25 × 10^-6 m Therefore, the uncertainty in energy is: ΔE = ħπΔc/Δλ = (1.055 × 10^-34 J ∙ s)(π)(3.00 × 10⁺⁸ m/s)/(2.25 × 10^-6 m) ΔE = 1.66 × 10^-18 J/photon Converting to eV, we get: ΔE = 1.66 × 10^-18 J/photon × (1 eV/1.602 × 10^-19 J) = 1.04 × 10⁺¹ eV/photon Rounding to two significant figures, we get ΔE ≈ 2 × 10^-5 eV/photon, which is closest to choice B.

Uncertainty Principle Using ΔEΔt ≥ ħ: a Laser Produces a Beam