Problem — Spectral width and mean lifetime

Problem

An atom emits a photon in a time $\tau \approx 10$ ns (mean lifetime of the excited state). Estimate the relative width $\Delta f/f$ of the emitted line around $\lambda = 500$ nm using the time-energy uncertainty relation $\Delta E \sim \hbar/\tau$.

Solution

Step 1 — Time-energy uncertainty. A state that lives for a finite time $\tau$ does not have a perfectly defined energy:

$\Delta E \sim \frac{\hbar}{\tau} = \frac{1{,}055\cdot 10^{-34}}{10\cdot 10^{-9}} \approx 1{,}1\cdot 10^{-26}\;\text{J}$

Step 2 — Photon energy.

$E = \frac{hc}{\lambda} = \frac{1240}{500}\;\text{eV} \approx 2{,}48\;\text{eV} = 2{,}48\cdot 1{,}6\cdot 10^{-19} \approx 3{,}97\cdot 10^{-19}\;\text{J}$

Step 3 — Relative width. Since $E = hf$, the relative width in frequency equals that in energy:

$\frac{\Delta f}{f} = \frac{\Delta E}{E} = \frac{1{,}1\cdot 10^{-26}}{3{,}97\cdot 10^{-19}} \approx 2{,}7\cdot 10^{-8}$

$\ev{\frac{\Delta f}{f} \approx 2{,}7\cdot 10^{-8}}$

Comment. The line is not infinitely thin: it has an intrinsic width (natural broadening) due to the finite lifetime of the state. The value $\sim 10^{-8}$ is tiny — atomic lines are extremely sharp — but not zero. Longer-lived states (large $\tau$) give narrower lines, the principle underlying the precision of atomic clocks; short-lived states give broad lines.

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Topics: Fisica quantistica Concepts: Principio di indeterminazione · Fotone Skills: Impostazione simbolica · Conversione di unità