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

  • Thread starter Lebnm
  • Start date
31
1
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?
 

DrClaude

Mentor
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2,993
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.
 

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