Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

will yield for variations as close as possible to those predicted by the exact N-particle Schrodinger equation. As the one-particle potential V₁ may also be time-dependent, it will be written as V₁(t).

1-a. Functional variation

      Let us introduce the functional of |Ψ(t)〉:

(4)

It can be shown that this functional is stationary when |Ψ(t)〉 is solution of the exact Schrodinger equation (an explicit demonstration of this property is given in § 2 of Complement F_XV. If |Ψ(t)〉 belongs to a variational family, imposing the stationarity of this functional allows selecting, among all the family kets, the one closest to the exact solution of the Schrodinger equation. We shall therefore try and make this functional stationary, choosing as the variational family the set of kets written as in (1) where the individual ket |θ(t)〉 is time-dependent.

As condition (3) means that the norm of remains constant, the second bracket in expression (4) must be zero. We now have to evaluate the average value of the Hamiltonian H(t) that, actually, has been already computed in (34) of Complement C_XV:

      (5)

      The only term left to be computed in (4) contains the time derivative.

      This term includes the diagonal matrix element:

(6)

      For an infinitesimal time dt, the operator is proportional to the difference , hence to the difference between two creation operators associated with two slightly different orthonormal bases. Now, for bosons, all the creation operators commute with each other, regardless of their associated basis. Therefore, in each term of the summation over k, we can move the derivative of the operator to the far right, and obtain the same result, whatever the value of k. The summation is therefore equal to N times the expression:

(7)

      Now, we know that:

      (8)

Using in (6) the bra associated with that expression, multiplied by N, we get:

      (9)

      Regrouping all these results, we finally obtain:

      (10)

1-b. Variational computation: the time-dependent Gross-Pitaevskii equation

      We now make an infinitesimal variation of |θ(t)〉:

      (11)

      in order to find the kets |θ(t)〉 for which the previous expression will be stationary. As in the search for a stationary state in Complement C_XV, we get variations coming from the infinitesimal ket eiχ |δθ(t)〉 and others from the infinitesimal bra e^–iχ 〈δθ(t)|; as χ is chosen arbitrarily, the same argument as before leads us to conclude that each of these variations must be zero. Writing only the variation associated with the infinitesimal bra, we see that the stationarity condition requires |θ(t)〉 to be a solution of the following equation, written for |φ(t)〉:

(12)

The mean field operator is defined as in relations (45) and (46) of Complement C_XV by a partial trace:

(13)

      where P^φ(t) is the projector onto the ket |φ(t)〉:

      (14)

As we take the trace over particle 2 whose state is time-dependent, the mean field is also time-dependent. Relation (12) is the general form of the time-dependent Gross-Pitaevskii equation.

      Let us return, as in § 2 of Complement C_XV, to the simple case of spinless bosons, interacting through a contact potential:

(15)

Using definition (13) of the Gross-Pitaevskii potential, we can compute its effect in the position representation, as in Complement C_XV. The same calculations as in §§ 2-b-β and 2-b-ϒ of that complement allow showing that relation (12) becomes the Gross-Pitaevskii time-dependent equation (N is supposed to be large enough to permit replacing N — 1 by N):

(16)

      Normalizing the wave function φ(r, t) to N:

(17)