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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)

equation (16) simply becomes:

(18)

Comment:

It can be shown that this time evolution does conserve the norm of |φ(t)〉, as required by (3). Without the nonlinear term of (16), it would be obvious since the usual Schrödinger equation conserves the norm. With the nonlinear term present, it will be shown in § 2-a that the norm is still conserved.