We had mentioned that from the partition function \(Z\), all possible thermodynamical quantities can be determined. In this problem, we will use the harmonic oscillator's partition function of Eq. (12.147) to do some calculations.

21 M. Planck, "Zur Quantenstatistik des Bohrschen Atommodells," Ann. Phys. 380(23), 673-684 (1924);

A. I. Larkin, "Thermodynamic functions of a low-temperature plasma," J. Exptl. Theoret. Phys.

(U.S.S.R.) 38, 1896-1898 (1960), Sov. Phys. JETP 11(6), 1363-1364 (1960).

22 C. A. Rouse, "Comments on the Planck-Larkin partition function," Astrophys. J. 272, 377-379 (1983).

(a) Show that the average energy per particle \(\langle Eangle\) of a system in thermal equilibrium can be determined by
\[\begin{equation*}\langle Eangle=-\frac{d \log Z}{d \beta} \tag{12.152}\end{equation*}\]
What is the average energy of a particle in the harmonic oscillator?
(b) Show that the entropy \(S\) of a system in thermal equilibrium can be expressed as
\[\begin{equation*}S=-\sum_{n} p_{n} \log p_{n}=\beta\langle Eangle+\log Z \tag{12.153}\end{equation*}\]
What is the entropy of the harmonic oscillator? Does the entropy in the temperature to zero limit make sense physically?
(c) What if we used the path integral in these expressions instead of the partition function? What would \(\beta\) be in that case? Do these expressions make sense? Test it out for a free particle.