# An Operator Identity relating to Trace [duplicate]

Suppose that $\hat H$ is an operator (typically a Hamiltonian) and $\beta$ is a positive parameter (typically $\beta=1/k_BT$). Show that $$\mathbf{Tr}\Big(e^{-\beta\hat H}\Big) \geq \sum_{k}e^{-\beta\langle k\vert\hat H k\rangle}$$ My first step is using a Taylor expansion to yield $$\sum_{n}\frac{(-\beta)^n}{n!}\sum_{k}\Big(\langle k\vert\hat H^n\vert k\rangle-\big(\langle k\vert\hat H k\rangle\big)^n\Big)\geq 0$$ How can I move forward?
• What happens if $\{\vert k\rangle\}$ is an eigenbasis of $\hat{H}$? – Nephente Oct 2 '14 at 6:09
• @nephente: If ${\vert k \rangle}$ happen to be eigenbasis of $\hat H$, the "=" holds. – Roger209 Oct 2 '14 at 6:27