The Planck radiation law explains how a blackbody emits electromagnetic radiation at a specific temperature, based on the assumption that the energy of each oscillator in the body can only have discrete values.

In June 1900, Lord Rayleigh published the Rayleigh-Jeans law, which is now known as a flawed attempt in physics to describe the spectral radiance of electromagnetic radiation as a function of wavelength from a blackbody at a given temperature. The mistake that he made was to use the equipartition theorem to assume that each oscillation mode within a blackbody has an average energy of . In December of the same year, the German physicist Max Planck presented the Planck radiation law, which assumed that the energy of an oscillator of frequency came in discrete bundles:

where and is a proportionality constant called the ** Planck constant**.

According to the Boltzmann distribution, the probability of a mode with frequency associated with the state is

The average energy of the mode of frequency is

Let .

Substituting the Taylor series of and in the above equation,

Substituting eq4 in eq2,

which is the mathematical expression of the Planck distribution law.

###### Question

Show that in the classical limit, the average energy of a mode in eq4 is consistent with the equipartition theorem.

###### Answer

In the classical limit, and we can expand as the Taylor series . Substituting the series in eq4 and ignoring the higher powers of the series because , we have .