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

G.R.Dixon, 12/27/2006

Given a unit volume cavity filled with black body radiation at temperature T, Planck’s formula for the radiant energy in (l,dl) is

. (1)

At T=1500, Eq.1 can be numerically integrated to determine that

. (2)

Alternatively one can use Eq. 1 and the Wien displacement law to derive the formula

. (3)

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/l^{4}. 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:

. (4)

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

. (5)

Here n(l,T)dl is the number of photons in (l,dl) at temperature T. More explicitly,

. (6)

And Eq. 6 can also be numerically integrated to find that

. (7)

We can divide E_{total} by n_{total} to find <e>, the average photon energy:

. (8)

Division of this result by kT produces

. (9)

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

. (10)

We can check the generality of Eq. 3 by again numerically integrating the integral in Eq. 2. The result confirms Eq. 10:

. (11)

We similarly find that

(12)

and

. (13)

Thus at the higher temperature there are more photons, with greater average energies (or greater average frequencies). But note that here again

. (14)

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.