D-Dimensional Blackbody Cavity

Consider a D-dimensional hypercube blackbody cavity. What is the energy density as a function of temperature? It is not necessary to derive the multiplicative constant. Assume that the radiation is in quanta of energy E(ω) = hω.

SOLUTION

For a photon gas the average number of photons per mode ω is given by

(1)

The energy

(2)

where V is the volume of the hypercube:

Substituting x = hω/τ into (2), we obtain

(3)

The energy density is simply

(4)

For D = 3 we recover the Stefan–Boltzmann law: