I Expectation value of the occupation number in the FD and BE distributions

Lebnm

In the derivation of the Fermi-Dirac and Bose-Einstein distributions, we compute the Grand Partion Function $Q$. With $Q$, we can compute the espection value of the occupation number $n_{l}$. This is the number of particles in the same energy level $\varepsilon _{l}$. The book I am reading write $$\langle n_{l} \rangle = - \frac{1}{\beta} \frac{\partial \ }{\partial \varepsilon _{l}}\ ln \ Q,$$ but the book dosen't deduce this equation. I know that the expection value of the number operator $\hat{N}$ is given by $$\langle \hat{N} \rangle = z \frac{\partial \ }{\partial z}\ ln \ Q,$$ where $z = e^{\beta \mu}$ ( $\mu$ is the chemical potential, and I am using the Grand Canonical Ensemble), and that the eigenvalues of $\hat{N}$ are $N = \sum _{l} n_{l}$. How the first equation follow from these relations?

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DrClaude

Mentor
The grand partition function is given by
$$Q = \sum_l e^{- \beta n_l (\varepsilon_l - \mu)}$$
therefore
\begin{align*} -\frac{1}{\beta} \frac{\partial}{\partial \varepsilon_l} \ln Q &= -\frac{1}{\beta} \left[ \frac{1}{Q} \frac{\partial Q}{\partial \varepsilon_l} \right] \\ &= -\frac{1}{\beta} \left[ \frac{1}{Q} \sum_l (- \beta n_l ) e^{- \beta n_l (\varepsilon_l - \mu)} \right] \\ &= \frac{1}{Q} \sum_l n_l e^{- \beta n_l (\varepsilon_l - \mu)} \\ &= \langle n_l \rangle \end{align*}
where the last equality is obtained from the formula for calculating expectation values.

• Demystifier

"Expectation value of the occupation number in the FD and BE distributions"

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