 Solution

Treat the ions as a stationary neutralizing background and displace the cold electrons by a small longitudinal distance $\xi$. Their electric polarization is $P=-n_{e}e\xi$, so the restoring electric field is $E=-P/{ϵ}_0=n_{e}e\xi /{ϵ}_0$. The electron equation of motion isFor $E=E_0\cos {\omega }_{\mathrm{pe}}t$, the oscillatory velocity is $v=-e{E}_0\sin {\omega }_{\mathrm{pe}}t/(m_e{\omega }_{\mathrm{pe}})$. HenceThe longitudinal mode exchanges electron kinetic energy and electrostatic field energy like a harmonic oscillator, whose mean kinetic and potential energies are equal over a cycle. This is a time-average statement for the coherent oscillation, not thermal equipartition with $k_{\mathrm{B}}T$.