Show that linearizing the Gross–Pitaevskii equation leads to the Bogoliubov energy dispersion for excitations. To do this,

(a) Write the wave function of the condensate as ψ₀ +δψ, where ψ = √n₀ e^{−iω₀ t} is the ground state wave function and δψ is a perturbation, and derive the linearized Gross–Pitaevskii equation by keeping terms no higher than linear in δψ or δψ^∗.

(b) Substitute into this equation the guess

(the ground state phase, modulated by a general plane wave state), and write a 2 × 2 matrix equation for α and β. Solve this equation for the eigenvalues and eigenvectors and show that your solution gives you the Bogoliubov spectrum.

(c) Show that the solution in the limit k → 0 corresponds to a density modulation of the condensate.

Transcribed Image Text:

8y =e-iwot (αei(kx-wt) + Be-i(kx-wt))