# Observables in Statistical Physics¶

## Observables in Equilibrium State¶

In statistics, the average of an observable is

$O = Tr(\hat O \rho)$

where $$\rho$$ is the “probability matrix”. In QM, this is

$\rho = \frac{ e^{-\beta H} }{\mathrm{Tr} ( e^{-\beta H} ) }$

If a system of QM comes to equilibrium, that means

1. $$\rho = \frac{ e^{-\beta H} }{\mathrm{Tr} e^{-\beta H} }$$;
2. $$\rho$$ is diagonal in energy eigen space.

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