Thermodynamics of the Harmonic Oscillator: Wien's Displacement Law and the Planck Spectrum

A thermodynamic analysis of the harmonic oscillator is presented. Motivation for the study is provided by the blackbody radiation spectrum; when blackbody radiation is regarded as a system of noninteracting harmonic oscillator modes, the thermodynamics follows from that of the harmonic oscillators. Using the behavior of a harmonic oscillator thermodynamic system under a quasi-static change of oscillator frequency w, we show that the thermodynamic functions can all be derived from a single function of w/T, analogous to Wien's displacement theorem. The high- and low-frequency energy limits allow asymptotic energy forms involving T alone or w alone, corresponding to energy equipartition and zero-point energy. It is noted that the Planck spectrum with zero-point radiation corresponds to the function satisfying the Wien displacement result which provides the smoothest possible interpolation between energy equipartition at low frequency and zero-point energy at high frequency.