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[Statistical Physics] Probability of finding # photons in the mode

  1. Apr 2, 2014 #1
    1. The problem statement, all variables and given/known data

    A cavity contains black body radiation at temperature T = 500 K. Consider an optical mode in the cavity with frequency ω=2.5x10[itex]^{13}[/itex] Hz. Calculate

    a) the probability of finding 0 photons in the mode
    b) the probability of finding 1 photon in the mode
    c) the mean number of photons in the mode.


    2. Relevant equations

    Possibly <n> = [itex]\frac{1}{exp(\frac{\hbar \omega}{k_{b} T})}[/itex]

    3. The attempt at a solution

    Plugging the numbers into the equation above gives the answer to c (I think), which comes out to 1.855. However I thought that you could only have 0 or 1 photons in a given mode.

    Not sure how to go about a) and b).
     
  2. jcsd
  3. Apr 2, 2014 #2
    Photons are bosons. Bosons do not follow Pauli's exclusion principle. Any number (from zero to infinity) of photons may occupy any given mode.
     
  4. Apr 3, 2014 #3
    Ah ok, so the mean number of photons might still be ok.

    How do I go about answering a and b?
     
  5. Apr 3, 2014 #4

    BvU

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    Just out of curiosity: how can you get <n> > 1 if all the factors in the exponential are > 0 ?
     
  6. Apr 3, 2014 #5
    You've just made me realise the equation should have a -1 on the bottom. Can't edit my original post for some reason.
     
  7. Apr 3, 2014 #6
    Ok think I've got the relevant equation for a) and b) now.

    For any future readers:

    P(n) = [itex]\frac{1 - exp(-\frac{ħw}{kT})}{exp(\frac{nħw}{kT})}[/itex]

    where n is the number of photons in the mode.
     
    Last edited: Apr 3, 2014
  8. Apr 3, 2014 #7

    TSny

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    Note that in your expression P(n) → ∞ as n→∞. So, it can't be correct since a probability can't be greater than 1.

    Maybe you need to switch the numerator and denominator.
     
  9. Apr 3, 2014 #8
    Ooh thanks for pointing that out, I accidentally put a minus in there.
     
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