Generally, material physics of finite (i.e. non-zero) temperature derives its macroscopic relations from small scale considerations. All observables are essentially determined by the relation between the possible microscopic states and their energy, which makes evaluation of the partition function possible. This is why the computation of energy levels $\varepsilon_r$ or dispersion relations $\hbar\omega({\bf k})$ are of central importance. Quantum mechanically, this requires computing eigenvalues of the Hamiltonian.
todo: Ensemble average
$\langle A\rangle_H := \dfrac{\mathrm{tr}(\mathrm e^{-\beta H}A)}{\mathrm{tr}(\mathrm e^{-\beta H})} = \dfrac{1}{Z(\beta)} \mathrm{tr}({\mathrm e}^{-\beta H}A) $