A simple harmonic one-dimensional oscillator has energy levels $E_n=$ $(n+1 / 2) \hbar \omega$, where $\omega$ is the characteristic oscillator (angular) frequency and $n=0,1,2, \ldots$.
(a) Suppose the oscillator is in thermal contact with a heat reservoir at temperature $T$, with $\frac{k T}{\hbar \omega} \ll 1$. Find the mean energy of the oscillator as a function of the temperature $T$.
(b) For a two-dimensional oscillator, $n=n_x+n_y$, where
$$
E_{n_x}=\left(n_x+\frac{1}{2}\right) \hbar \omega_x, \quad E_{n_y}=\left(n_y+\frac{1}{2}\right) \hbar \omega_y,
$$
$n_x=0,1,2, \ldots$ and $n_y=0,1,2, \ldots$, what is the partition function for this case for any value of temperature? Reduce it to the degenerate case $\omega_x=\omega_y$.
(c) If a one-dimensional classical anharmonic oscillator has potential energy $V(x)=c x^2-g x^3$, where $g x^3 \& c x^2$, at equilibrium temperature $T$, carry out the calculations as far as you can and give expressions as functions of temperature for
1) the heat capacity per oscillator and
2) the mean value of the position $x$ of the oscillator.