Is this the correct way to estimate the natural lifetime of an atomic state?

[doomhalo]

Hi there,

I've just been having a little trouble with this short question from a past exam paper...

Homework Statement

"An atomic state has a dominant decay mode which produces an emission line of wavelength $6 \times 10^{-7} m$ and natural width $10^{-13} m$. Estimate it's natural lifetime.

Homework Equations

$$\Delta E \Delta T \geq \frac{\hbar}{2}$$
$$E = \frac{hc}{\lambda}$$

The Attempt at a Solution

$$\tau \approx \frac{\hbar}{2E}$$
$$\tau \approx \frac{\lambda}{4\pi c} \approx 1.59 \times 10^{-16}$$

I was just wondering if this seemed right? I'm concerned that I've not used the natural width provided in the question but I'm not sure whether it's a case of I've a) Missed a relevant equation, or b) I've used the equations I do have wrong.

Thank you in advance!

 

[unscientific]

The mistake here is that you used:$$\Delta E = \Delta \lambda$$

What is $\Delta E$ in terms of $\lambda$ and $\Delta \lambda$?

 

[doomhalo]

The mistake here is that you used:$$\Delta E = \Delta \lambda$$

What is $\Delta E$ in terms of $\lambda$ and $\Delta \lambda$?

Ah I see, so
$$\Delta E = \frac{hc}{\lambda^2} \Delta\lambda$$

And then

$$\tau \approx \frac{\lambda^2}{4\pi c \Delta\lambda}$$

?

Thank you very much!

 

[unscientific]

Yes that's right. That gives a reasonable answer. (0.3s I think)