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

1. Apr 2, 2014

Flucky

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$^{13}$ 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> = $\frac{1}{exp(\frac{\hbar \omega}{k_{b} T})}$

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. Apr 2, 2014

dauto

Photons are bosons. Bosons do not follow Pauli's exclusion principle. Any number (from zero to infinity) of photons may occupy any given mode.

3. Apr 3, 2014

Flucky

Ah ok, so the mean number of photons might still be ok.

4. Apr 3, 2014

BvU

Just out of curiosity: how can you get <n> > 1 if all the factors in the exponential are > 0 ?

5. Apr 3, 2014

Flucky

You've just made me realise the equation should have a -1 on the bottom. Can't edit my original post for some reason.

6. Apr 3, 2014

Flucky

Ok think I've got the relevant equation for a) and b) now.

P(n) = $\frac{1 - exp(-\frac{ħw}{kT})}{exp(\frac{nħw}{kT})}$

where n is the number of photons in the mode.

Last edited: Apr 3, 2014
7. Apr 3, 2014

TSny

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.

8. Apr 3, 2014

Flucky

Ooh thanks for pointing that out, I accidentally put a minus in there.