The energy levels of a quantum-mechanical, one-dimensional, anharmonic oscillator may be approximated as

\[
\varepsilon_{n}=\left(n+\frac{1}{2}ight) \hbar \omega-x\left(n+\frac{1}{2}ight)^{2} \hbar \omega ; \quad n=0,1,2, \ldots
\]

The parameter \(x\), usually \(\ll 1\), represents the degree of anharmonicity. Show that, to the first order in \(x\) and the fourth order in \(u(\equiv \hbar \omega / k T)\), the specific heat of a system of \(N\) such oscillators is given by

\[
C=N k\left[\left(1-\frac{1}{12} u^{2}+\frac{1}{240} u^{4}ight)+4 x\left(\frac{1}{u}+\frac{1}{80} u^{3}ight)ight] .
\]

The correction term here increases with temperature.