Author's Note: This article has been revised, but the end results, Eq. 9 and Eq. 14, appear to be correct. A more complete form for the constant is 2.700768682. This result is the same as that given in Eq. 1.21 of Introduction To Quantum Mechanics, Bransden & Joachain, ISBN 0-582-44498-5.
On the Temperature Independence of Energy Partitioning Among Photons in Black Body Radiation
Given a unit volume cavity filled with black body radiation at temperature T, Planck’s formula for the radiant energy in (l,dl) is
At T=1500, Eq.1 can be numerically integrated to determine that
Alternatively one can use Eq. 1 and the Wien displacement law to derive the formula
Historically, Rayleigh and Jeans deduced that the maximum number of standing waves (a.k.a. "modes") that could exist in a unit volume is 8p/l4. Later, Einstein suggested that the energy of a photon is always hc/l. These results suggest that Eq. 1 be "parsed" in the following manner:
In this article we shall interpret Eq. 4 as follows: The energy in (l,dl) equals the energy of a photon of wavelength l times the number of photons present at temperature T. Note here the postulate that only one photon occupies any given mode, and that all possible modes are not populated by photons. That is, Planck’s factor of 1/(exp(hc/kTl)-1) throttles back the number of modes occupied by photons (particularly at the smaller wavelengths).
Given this interpretation of Eq. 4, we can write the equation more compactly as
Here n(l,T)dl is the number of photons in (l,dl) at temperature T. More explicitly,
And Eq. 6 can also be numerically integrated to find that
We can divide Etotal by ntotal to find <e>, the average photon energy:
Division of this result by kT produces
Thus at T=1500 the average photon energy is 2.7kT.
Let us now consider a hotter cavity, say one at temperature T=5000. Assuming Eq. 3 is general produces
We can check the generality of Eq. 3 by again numerically integrating the integral in Eq. 2. The result confirms Eq. 10:
We similarly find that
Thus at the higher temperature there are more photons, with greater average energies (or greater average frequencies). But note that here again
This is the same as the equipartition factor found at T=1500 (Eq. 9). Thus the result that <e>=2.7kT is true at both T=1500 and T=5000, and like Eq. 3 is probably generally true.