1. May 8, 2014

doomhalo

Hi there,

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

1. The problem statement, all variables and given/known data
"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.

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

3. 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.

2. May 9, 2014

unscientific

The mistake here is that you used:

$$\Delta E = \Delta \lambda$$

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

3. May 9, 2014

doomhalo

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!

4. May 9, 2014

unscientific

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