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

 

[paranormal]

To solve for part (a) and (b), you can use the Boltzmann distribution, which relates the probability of finding a certain number of particles in a system to the energy of that state. In this case, the energy of a photon in the optical mode can be calculated using the Planck's law equation:
E = h*w, where h is Planck's constant and w is the frequency of the mode.

(a) To find the probability of finding 0 photons in the mode, you can use the Boltzmann distribution:
P(0 photons) = e^(-E/kT) = e^(-h*w/kT)

(b) To find the probability of finding 1 photon in the mode, you can use the Boltzmann distribution:
P(1 photon) = e^(-E/kT) = e^(-h*w/kT)

(c) To find the mean number of photons in the mode, you can use the average energy formula:
< E > = (sum of all possible energies * probability of each energy)
< E > = (0 * P(0 photons)) + (h*w * P(1 photon))
< E > = h*w * P(1 photon)

Therefore, the mean number of photons in the mode is equal to P(1 photon). You can use the equations from parts (a) and (b) to calculate the mean number of photons in the mode.