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 $\overline{E}=kT$ . 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 $\nu$ came in discrete bundles:

$E_{n,\nu}=nh\nu$

where $n=0,1,2,\cdots$ and $h$ is a proportionality constant called the Planck constant.

According to the Boltzmann distribution, the probability of a mode $P_{n,\nu}$ with frequency $\nu$ associated with the state $E_{n,\nu}$ is

$P_{n,\nu}=\frac{e^{-\frac{E_{n,\nu}}{kT}}}{\sum_{n=0}^{\infty}e^{-\frac{E_{n,\nu}}{kT}}}$

The average energy $\overline{E}$ of the mode of frequency $\nu$ is

$\overline{E}=\sum_{n=0}^{\infty}E_{n,\nu}p_{n,\nu}=\frac{\sum_{n=0}^{\infty}E_{n,\nu}e^{-\frac{E_{n,\nu}}{kT}}}{\sum_{n=0}^{\infty}e^{-\frac{E_{n,\nu}}{kT}}}=h\nu\frac{\sum_{n=0}^{\infty}ne^{-\frac{nh\nu}{kT}}}{\sum_{n=0}^{\infty}e^{-\frac{n\nu}{kT}}}$

Let ${x=e^{-\frac{h\nu}{kT}}}$ .

$\overline{E}=h\nu\frac{\sum_{n=0}^{\infty}nx^n}{\sum_{n=0}^{\infty}x^n}=h\nu\biggr$\frac{x+2x^2+3x^3+\cdots}{1+x+x^2+\cdots}\biggr$=h\nu x\biggr$\frac{1+2x+3x^2+\cdots}{1+x+x^2+\cdots}\biggr$$

Substituting the Taylor series of ${\frac{1}{1-x}=1+x+x^2+\cdots}$ and ${\frac{1}{(1-x)^2}=1+2x+3x^2+\cdots}$ in the above equation gives

$\overline{E}=\frac{h\nu x}{1-x}=\frac{h\nu}{x^{-1}-1}=\frac{h\nu}{e^{\frac{h\nu}{kT}}-1}\;\;\;\;\;\;\;\;4$

Substituting eq4 in eq2 yields

$u(\nu)d\nu=\frac{8\pi h\nu^3}{c^3}\frac{1}{e^{\frac{h\nu}{kT}}-1}d\nu\;\;\;\;\;\;\;\;5$

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, ${h\nu\rightarrow 0}$ and we can expand ${e^\frac{h\nu}{kT}}$ as the Taylor series ${e^\frac{h\nu}{kT}=1+\frac{h\nu}{kT}+\frac{1}{2!}\biggr$\frac{h\nu}{kT}\biggr$^2+\cdots}$ . Substituting the series in eq4 and ignoring the higher powers of the series because ${\frac{h\nu}{kT}\rightarrow 0}$ , we have $\overline{E}=kT$ .

Planck radiation law

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