Canonical and Microcanonical Ensembles

Exercise 7: (A) Use Eqs. (36) and (11) to show that in a canonical ensemble the probability $p_j$ of observing the system in quantum state $\vert j>$, where

$$H \vert j> = E_j \vert j>,$$ (43)

is the Boltzmann probability distribution

$$p_j = Z^{-1} exp(-\beta E_j) = exp(-\beta(E_j-A)),$$ (44)

where $\beta=(k T)^{-1}$, with $T$ the temperature of the ensemble and $k$ the Boltzmann constant. (B) Show that for a microcanonical ensemble, where all of the states $\vert j>$ have the same energy $E_j=E$, the probability of observing the system in state $\vert j>$ is

$$p_j=\frac{1}{\Omega},$$ (45)

where $\Omega$ is the total number of states. Note that $p_j$ is, therefore, independent of the particular state $\vert j>$ in a microcanonical ensemble. Note that according to Eqs. (23) and (45), the entropy of a microcanonical ensemble corresponds to the Boltzmann definition of entropy,

$$S = k \Omega.$$ (46)