Briefly describe the microcanonical, canonical and grand canonical ensembles. Why do they agree in the thermodynamic limit?

A harmonic oscillator in one spatial dimension has Hamiltonian

$$H=\frac{p^{2}}{2 m}+\frac{m}{2} \omega^{2} x^{2}$$

Here $p$ and $x$ are the momentum and position of the oscillator, $m$ is its mass and $\omega$ its frequency. The harmonic oscillator is placed in contact with a heat bath at temperature $T$. What is the relevant ensemble?

Treating the harmonic oscillator classically, compute the mean energy $\langle E\rangle$, the energy fluctuation $\Delta E^{2}$ and the heat capacity $C$.

Treating the harmonic oscillator quantum mechanically, compute the mean energy $\langle E\rangle$, the energy fluctuation $\Delta E^{2}$ and the heat capacity $C$.

In what limit of temperature do the classical and quantum results agree? Explain why they differ away from this limit. Describe an experiment for which this difference has implications.