This problem is similar to Problem 7 8 of the Bose gas and can be done the same way In the limit (z ightarrow infty ), which corresponds to (T ightarrow 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 ight) 1 2 For comparison...

This problem is similar to Problem 7.8 of the Bose gas and can be done the same

Question:

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 $w_{0}=u_{F} / \sqrt{3}$.