Show that, in the case of a degenerate gas of fermions \(\left(T \ll T_{F}ight.\) ), the correlation function \(g(r)\), for \(r \gg \hbar / p_{F}\), reduces to the expression

\[
g(r)-1=-\frac{3(m k T)^{2}}{4 p_{F}^{3} \hbar r^{2}}\left\{\sinh \left(\frac{\pi m k T r}{p_{F} \hbar}ight)ight\}^{-2} .
\]

Note that, as \(T ightarrow 0\), this expression tends to the limiting form

\[
g(r)-1=-\frac{3 \hbar}{4 \pi^{2} p_{F} r^{4}} \propto \frac{1}{r^{4}}
\]
\footnotetext{ \({ }^{14}\) Note that, in the classical limit \((\hbar ightarrow 0)\), the infinitely rapid oscillations of the factor \(\exp \{i(\boldsymbol{p} \cdot \boldsymbol{r}) / \hbar\}\) make the integral vanish. Consequently, for an ideal classical gas, the function \(g(r)\) is identically equal to 1 . Quantum-mechanical systems of identical particles exhibit spatial correlations due to Bose and Fermi statistics even in the absence of interactions. It is not difficult to see that, for \(n \lambda^{3} \ll 1\) where \(\lambda=h / \sqrt{(2 \pi m k T)}\),
}

\[
g(r) \simeq 1 \pm \frac{1}{g_{s}} \exp \left(-2 \pi r^{2} / \lambda^{2}ight)
\]

compare with equation (5.5.27).