Uncertainty time/frequency

[Gavroy]

hi

i am looking for the correct equation, that gives me the linewidth by knowing the time of the emission process.

so far, i found in the internet:

Δf=1/(Δt), Δf=1/(4πΔt) and Δf=1/(2Δt)

can you tell me, when i have to use which of these equations and maybe whether you think that these equations are correct?

sry for my english, still practising! ;-)

 

[Bill_K]

Gavroy, Getting the factor right requires saying precisely what you mean by the lifetime and the linewidth. A decaying state is described by a wavefunction |ψ|² ~ exp(-Γt/ħ), so the lifetime of the state may be defined as Δτ = ħ/Γ.

The shape of the line is a Lorentzian, 1/((E-E₀)² + (Γ/2)²) which reaches its half-height at E = E₀ ± Γ/2, so the "width" in that sense is ΔE = Γ. All right now ΔE = ħ Δω = 2π ħ Δf, so putting it all together you get Δf = 1/(2π Δτ).