What are the eigenvalues of a state in thermal equilibrium?

Question

Suppose the density matrix $\rho$ with eigenvalues $k_{i}$.

Now consider the density matrix $\rho$ in a thermal equilibrium with temperature $T$. Let's show the density matrix with $\rho(T)$ in this case.

What are the eigenvalues of $\rho(T)$?

Answer 1

Thermal equilibrium Gibbs state is usually defined as $$\rho = \frac{1}{Z} e^{-\beta H} = \frac{1}{Z} \text{diag}(e^{-\beta E_0}, e^{-\beta E_1} \cdots)\,, $$ where $\beta$ is the inverse temperature defined as $$ \beta = \frac{1}{K_B T}\,,$$

And $Z$ is the partition function $$ Z = \sum_i e^{-\beta E_i} \,.$$

So, the eigenvalues $\{ \lambda_i \}$ are $$ \lambda_i = \frac{e^{-\beta E_i}}{Z}\,. $$