Discussion
Average energy of a Planck's oscillator of frequency v is :
Answer: C
The average energy is \(E = \frac{\hbar v}{e^{\hbar v/kT} - 1}\).
Planck assumed the oscillator energy is quantised in steps of \(h\nu\) (the paper writes \(\hbar v\) for the quantum). Weighting each level \(nh\nu\) with the Boltzmann factor gives the mean energy:
\[ \begin{aligned} \langle E \rangle &= \frac{\sum nh\nu\,e^{-nh\nu/kT}}{\sum e^{-nh\nu/kT}} \\ &= \frac{h\nu}{e^{h\nu/kT} - 1} \end{aligned} \]
A single quantum or \(n\) quanta is the energy of a particular state, not the thermal average, and \(mc^{2}\) is rest energy. At high temperature the expression tends to \(kT\), the classical value.
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