Chapter 2, Problem 2.63P

(a)

To determine

The thermal average of the system’s energy can be written as $\overline{E}={\displaystyle \sum _n{E}_{n}P\left(n\right)}=-\frac{\partial }{\partial \beta }\ln \left(Z\right)$.

(b)

To determine

To show that the partition function Z is $Z=\frac{e^{-\beta \hslash \omega /2}}{1-e^{-\beta \hslash \omega }}$

(c)

To determine

To show that for quantum oscillator $\overline{E}=\left(\frac{\hslash \omega }{2}\right)\frac{1+e^{-\beta \hslash \omega }}{1-e^{-\beta \hslash \omega }}$

(d)

To determine

To show that $C=3k_B{\left(\frac{{\theta }_E}{T}\right)}^2\frac{e^{{\theta }_E/T}}{{\left(e^{{\theta }_E/T}-1\right)}^2}$

(e)

To determine

Sketch the graph of $C/k_B$ versus $T/{\theta }_E$