Planck radiation law

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.



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


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 .


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