Statistical Physics - blackbody radiation

[Matt atkinson]

Homework Statement

A cavity contains black body radiation at temperature at T=500K. Consider a optical mode in the cavity with frequency w=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.

Homework Equations

The Attempt at a Solution

Okay so I'm not sure where to start basically with part (a) and (b), but I made an attempt at part (c)
I used the equation;
\bar{n}=\frac{1}{e^{\frac{\hbar\omega}{k_b T}}-1}=2.51\times 10^{-17} Photons
would really love a nudge in the right way, I've just gone blank on probability it's been so long since I last did it.

 

[Matt atkinson]

Bump! ;D

 

[az_lender]

A cavity contains black body radiation at temperature at T=500K. Consider a optical mode in the cavity with frequency w=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.

The Attempt at a Solution

Okay so I'm not sure where to start basically with part (a) and (b), but I made an attempt at part (c)
I used the equation;
\bar{n}=\frac{1}{e^{\frac{\hbar\omega}{k_b T}}-1}=2.51\times 10^{-17} Photons
would really love a nudge in the right way, I've just gone blank on probability it's been so long since I last did it.

The probability distribution will be a Poisson dstribution, so
P(X=n) = λⁿ e^-λ/n!
where λ is the mean of the distribution.
If your answer to (c) is correct, then (a) and (b) are easy,
just put in n=0 and n=1.
BTW if your formula for (c) is correct,
then your numerical answer for (c) is incorrect.

 

[TSny]

I used the equation;
\bar{n}=\frac{1}{e^{\frac{\hbar\omega}{k_b T}}-1}=2.51\times 10^{-17} Photons

The equation is correct, but your numerical result is incorrect. I get roughly 0.1. You should recheck the calculation.

For information on calculating the probability that the mode contains n photons, see for example the following discussion

http://physics.ucsc.edu/~drip/5D/photons/photons.pdf