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The result is known as the Planck distribution function and gives us the average number of photons in the mode with frequency σ.
From the Planck distribution function, we can derive the energy density in the cavity. Convince yourself that the total energy in the cavity is given by:
< 
>
Each 
corresponds to the energy at a particular frequency
σn, and summing over all of the averages should yield the total
energy. More explicitly:
< > = < s > 
σ = 
Here, we can use the standard quantum method of letting the cavity be a cube and quantizing the frequencies to obtain σn = nΠc/L if L is the length of a side of the cube.
We need one more trick to complete the derivation. The sum over
positive n in 3 dimensions becomes 
4Πn2 dn.
With those tools, we can plug through some more algebra to obtain:
= 
τ4
The result is known as the Stefan-Boltzmann law of radiation. The significant aspect of the formula is that the energy density is proportional to the fourth power of the temperature.
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