Bose-Einstein Statistics of Light Quanta

We can also try to deduce the radiation formula, not as above from the pure wave standpoint by quantisation of the cavity radiation, but from the standpoint of the theory of light quanta, that is to say, of a corpuscular theory. For this we must therefore develop the statistics of the light-quantum gas, and the obvious suggestion is to apply the methods of the classical Boltzmann statistics, as in the kinetic theory of gases the quantum hypothesis, introduced by Planck in his treatment of cavity radiation by the wave method, is of course taken care of from the first in the present case, in virtue of the fact that we are dealing with light quanta, that is, with particles (photons) with energy hv and momentum Av/c. It turns out, however, that the attempt to deduce Planck s radiation law on these lines also fails, as we proceed to explain. 

however, as we know from our previous investigations (p. 198), the only standing waves which can exist in a cubical cavity (of side a) are those which satisfy the conditions 

In the classical statistics ( 6, p. 9) the momentum space was divided into cells of arbitrary form and magnitudes co, the 

We now return to the statistics of light quanta, and begin with 

The Boltzmann law of distribution was obtained, let us repeat ( 6, p. 9), as the most probable distribution of the particles of a gas (in our case the light quantum gas) in the various sheets (called cells in our earlier investigation), subject to the two subsidiary conditions = n and = E when the number of particles and the total energy are given. For the distribution of energy in our light quantum gas we therefore find 

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