Electrostatic oscillations of electrons in a plasma: the simplest example of collective quantum-scale excitations
In an unmagnetized plasma, displaced electrons experience a restoring force from quasi-neutrality. The oscillation frequency at k=0 is the plasma frequency: ω_p = √(n₀e²/ε₀m_e). For finite wavelengths, thermal effects add a correction: ω² = ω_p² + 3k²v_th² — the Bohm-Gross dispersion relation. The group velocity v_g = dω/dk = 3kv_th²/ω > 0, so Langmuir waves are dispersive. For k·λ_D > 1 (λ_D = Debye length), waves are strongly Landau damped — electrons moving near the wave phase velocity absorb energy. The lower panel shows the full dispersion curve ω(k) with your selected k marked.