Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 1 / b / i / Solution

For $X=(x,p{)}^T$,Solving $A\sigma +\sigma A^T=b{b}^T$ givesThe canonical ensemble density proportional to $e^{-H/(k_{B}T)}$ for $H=p^2/(2m)+\lambda x^2/2$ factorizes into independent centered Gaussians. Its equipartition variances are exactly $⟨x^2⟩=k_{B}T/\lambda$ and $⟨p^2⟩=m{k}_{B}T$, while the absence of an $xp$ term gives ${\sigma }_{12}=0$.