Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / Section B / B2 / a

Standing-wave wave numbers are $k_n=n\pi /L$, with one positive mode index and two polarizations. Since $ϵ=\hslash ck$, the continuum density of states isThe factor two counts polarizations; counting independent positive and negative traveling momenta in addition would double count standing modes. The thermal energy isThus the one-dimensional photon-gas thermodynamics givesReplacing the mode sum by an integral requires the stated temperature to be large compared with the level-spacing scale. This is thermal energy above the vacuum contribution; a separate boundary-dependent vacuum energy is not included.