(12)
The mean field operator
where Pφ(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 CXV, to the simple case of spinless bosons, interacting through a contact potential:
Using definition (13) of the Gross-Pitaevskii potential, we can compute its effect in the position representation, as in Complement CXV. 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):
Normalizing the wave function φ(r, t) to N:
equation (16) simply becomes:
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.
1-c. Phonons and Bogolubov spectrum
Still dealing with spinless bosons, we consider a uniform system, at rest, of particles contained in a cubic box of edge length L. The external potential V1(r) is therefore zero inside the box and infinite outside. This potential may be accounted for by forcing the wave function to be zero at the walls. In many cases, it is however more convenient to use periodic boundary conditions (Complement CXIV, § 1-c), for which the wave function of the individual lowest energy state is simply a constant in the box. We thus consider a system in its ground state, whose Gross-Pitaevskii wave function is independent of r:
with a μ value that satisfies equation (16):
where n0 = N/L3 is the system density. Comparing this expression with relation (58) of Complement CXV allows us to identify μ with the ground state chemical potential. We assume in this section that the interactions between the particles are repulsive (see the comment at the end of the section):
α. Excitation propagation
Let us see which excitations can propagate in this physical system, whose wave function is no longer the function (19), uniform in space. We assume:
(22)
where δφ(r, t) is sufficiently small to be treated to first order. Inserting this expression in the right-hand side of (16), and keeping only the first-order terms, we find in the interaction term the first-order expression:
(23)
We therefore get, to first-order:
(24)
which shows that the evolution of δφ(r, t) is coupled to that of δφ*(r, t). The complex conjugate equation can be written as:
(25)
We can make the time-dependent exponentials on the right-hand side disappear by defining:
This leads us to a differential equation with constant coefficients, which can be simply expressed in a matrix form:
(27)
