 Solution

For the canonical quantum Gibbs state,If $H|n⟩=E_n|n⟩$, its spectral form is $\rho =Z^{-1}{\sum }_n{e}^{-\beta E_n}|n⟩⟨n|$, with a sum over any degeneracy and an integral where a continuous spectrum needs one. It is Hermitian, positive and has unit trace, and it commutes with $H$. The formula presumes a well-defined finite partition function. At zero temperature a nondegenerate stable system approaches its ground-state projector; a degenerate ground space requires specifying the ensemble or preparation.