A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with E_x = shω, where s is a positive integer or zero, and ω is the classical frequency of the oscillator. We have chosen the zero of energy at the state s = 0

(a) Show that for a harmonic oscillator the free energy is

F = τ log[1 – exp(–hω/τ)]        (87)

Note that at high temperatures such that τ >> hω we may expand the argument of the logarithm to obtain F ≈ τ log(hω/τ)

(b) From (87) show that the entropy is

The entropy is shown in figure(a) and the heat capacity in figure(b).

Transcribed Image Text:

hw - log [1– exp(-)] hw exp 1.