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Homework Help: Atomic State Natural Lifetime

  1. May 8, 2014 #1
    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 [itex] 6 \times 10^{-7} m [/itex] and natural width [itex] 10^{-13} m [/itex]. Estimate it's natural lifetime.

    2. Relevant equations
    [tex] \Delta E \Delta T \geq \frac{\hbar}{2} [/tex]
    [tex] E = \frac{hc}{\lambda} [/tex]

    3. The attempt at a solution

    [tex] \tau \approx \frac{\hbar}{2E} [/tex]
    [tex] \tau \approx \frac{\lambda}{4\pi c} \approx 1.59 \times 10^{-16} [/tex]

    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!
  2. jcsd
  3. May 9, 2014 #2
    The mistake here is that you used:

    [tex]\Delta E = \Delta \lambda[/tex]

    What is ##\Delta E## in terms of ##\lambda## and ##\Delta \lambda##?
  4. May 9, 2014 #3
    Ah I see, so
    [tex] \Delta E = \frac{hc}{\lambda^2} \Delta\lambda [/tex]

    And then

    [tex] \tau \approx \frac{\lambda^2}{4\pi c \Delta\lambda} [/tex]


    Thank you very much!
  5. May 9, 2014 #4
    Yes that's right. That gives a reasonable answer. (0.3s I think)
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