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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