Uncertainty principle using ΔEΔt ≥ ħ/2: An ultraviolet source produces a monochromatic beam of 200-nm light. A shutter allows a pulse to pass that is 10,000 wavelengths long. The uncertainty in the energy of a photon in this pulse is closest to which of the following? (ħ = 1.055 × 10^-34 J ∙ s = 6.59 × 10^-16 eV ∙ s, c = 3.00 × 10⁸ m/s) A) 5 × 10^-7 eV B) 5 × 10^-6 eV C) 5 × 10^-5 eV D) 5 × 10^-4 eV E) 5 × 10^-3 eV

Question

Quizplus · Accepted Answer

First, calculate the time duration ($\Delta t$) of the pulse. The length of the pulse is 10,000 wavelengths, and the wavelength ($\lambda$) is 200 nm or $200 \times 10^{-9}$ m. The speed of light ($c$) is $3.00 \times 10^8$ m/s. The time duration is given by $\Delta t = \frac{\text{distance}}{\text{speed}} = \frac{10,000 \times 200 \times 10^{-9}\, \text{m}}{3.00 \times 10^8\, \text{m/s}}$. This simplifies to $\Delta t = \frac{2 \times 10^{-3}\, \text{s}}{3}$, which is approximately $6.67 \times 10^{-7}$ s. Using the uncertainty principle $\Delta E \Delta t \geq \frac{\hbar}{2}$, where $\hbar = 1.055 \times 10^{-34}$ J·s, we find $\Delta E \geq \frac{\hbar}{2\Delta t}$. Substituting the values, $\Delta E \geq \frac{1.055 \times 10^{-34}}{2 \times 6.67 \times 10^{-7}}$ J. Converting to eV using $1\, \text{J} = 6.24 \times 10^{18}\, \text{eV}$, we get $\Delta E \approx 5 \times 10^{-5}$ eV.

Uncertainty Principle Using ΔEΔt ≥ ħ/2: an Ultraviolet Source Produces