Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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= 0. Adding and subtracting the two relations shows that both coefficients δc₁ and δc₂ must be zero. In other words, we can impose to be zero as just θ^*(r) varies but not θ(r) - or the opposite¹.

       β. Stationary condition: Gross-Pitaevskii equation

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).

       ϒ. Zero-range potential

      The Gross-Pitaevskii equation is often written in conjunction with an approximation where the particle interaction potential has a microscopic range, very small compared to the distances over which the wave function φ(r) varies. We can then substitute:

      (24)

      where the constant g is called the “coupling constant”; such a potential is sometimes known as a “contact potential” or, in other contexts, a “Fermi potential”. We then get:

(25)

Whether in this form² or in its more general form (22), the equation includes a cubic term in φ(r). It may render the problem difficult to solve mathematically, but it also is the source of many interesting physical phenomena. This equation explains, for example, the existence of quantum vortices in superfluid liquid helium.

       δ. Other normalization

      Rather than normalizing the wave function φ(r) to 1 in the entire space, one sometimes chooses a normalization taking into account the particle number by setting:

      (26)

      This amounts to multiplying by the wave function we have used until now. At each point r of space, the particle (numerical) density n(r) is then given by:

      (27)

With this normalization, the factor (N — 1) in (25) is replaced by (N — 1)/N, which can generally be taken equal to 1 for large N. The Gross-Pitaevskii equation then becomes:

(28)

      As already mentioned, we shall see in § 4-a that μ is simply the chemical potential.

3. Generalization, Dirac notation

We now go back to the previous line of reasoning, but in a more general case where the bosons may have spins. The variational family is the set of the N-particle state vectors written in (7). The one-body potential may depend on the position r, and, at the same time, act on the spin (particles in a magnetic field gradient, for example).

3-a. Average energy

      To compute the average energy value , we use a basis {|θ_k〉} of the individual state space, whose first vector is |θ₁〉 = |θ〉.

      Using relation (B-12) of Chapter XV, we can write the average value as:

(29)

      Since is a Fock state whose only non-zero population is that of the state |θ₁〉, the ket

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