  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.
