This problem is similar to Problem 7.8 of the Bose gas and can be done the same
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This problem is similar to Problem 7.8 of the Bose gas and can be done the same way. In the limit \(z \rightarrow \infty\), which corresponds to \(T \rightarrow 0 K\),
\[
w^{2} \approx 2 k T \ln z / 3 m
\]
which tends to the limiting value \(2 \varepsilon_{F} / 3 m\). Thus
\[
w_{0}=\left(2 \varepsilon_{F} / 3 m\right)^{1 / 2}
\]
For comparison, the Fermi velocity \(u_{F}=\left(2 \varepsilon_{F} / m\right)^{1 / 2}\). It follows that \(w_{0}=u_{F} / \sqrt{3}\).
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