Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 84

We choose to impose the variation to be zero as only θ*(r) varies and for χ = 0. We must first add contributions coming from (13) and (14), then from (15). For this last contribution, we must add two terms, one coming from the variations due to θ*(r′), and the other from the variation due to θ*(r′). These two terms only differ by the notation in the integral variable and are thus equal: we just keep one and double it. We finally add the term due to the variation of the integral in (18), and we get:

(21)

This variation must be zero for any value of δf*(r); this requires the function that multiplies δf*(r) in the integral to be zero, and consequently that θ(r) be the solution of the following equation, written for φ(r):

(22)

This is the time-independent Gross-Pitaevskii equation. It is similar to an eigenvalue Schrodinger equation, but with a potential term:

(23)

which actually contains the wave function φ in the integral over d³r′; it is therefore a nonlinear equation. The physical meaning of the potential term in W₂ is simply that, in the mean field approximation, each particle moves in the mean potential created by all the others, each of them being described by the same wave function φ(r′); the factor (N — 1) corresponds to the fact that each particle interacts with (N — 1) other particles. The Gross-Pitaevskii equation is often used to describe the properties of a boson system in its ground state (Bose-Einstein condensate).