Integrating (7.6.17) by parts, show that the effective mass of an excitation, whose energymomentum relationship is denoted by \(\varepsilon(p)\), is given by

\[
m_{\mathrm{eff}}=\left\langle\frac{1}{3 p^{2}}\left\{\frac{d}{d p}\left(p^{4} \frac{d p}{d \varepsilon}ight)ight\}ightangle
\]

Check the validity of this result by considering the examples of (i) an ideal-gas particle, (ii) a phonon, and (iii) a roton.