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To deal with equation (39), we introduce the Gross-Pitaevskii operator , defined as a one-particle operator whose matrix elements in an arbitrary basis are {|u_i〉} given by:

(40)

which leads to:

(41)

where |v〉 and |v′〉 are two arbitrary one-particle kets – this can be shown by expanding these two kets on the basis {|u_i〉} and using relation (40). Note that this potential operator does not include an exchange term; this term does not exist when the two interacting particles are in the same individual quantum state. Equation (39) then becomes:

(42)

This stationarity condition must be verified for any value of the bra |δα〉, with only the constraint that it must be orthogonal to |θ〉 (according to relation (36)). This means that the ket resulting from the action of the operator on |θ〉 must have zero components on all the vectors orthogonal to |θ〉; its only non-zero component must be on the ket |θ〉 itself, which means it is necessarily proportional to |θ〉. In other words, |θ〉 must be an eigenvector of that operator, with eigenvalue μ (real since the operator is Hermitian):

(43)

We have just shown that the optimal value |φ〉 of |θ〉 is the solution of the Gross-Pitaevskii equation:

(44)

which is a generalization of (28) to particles with spin, and is valid for one- or two-body arbitrary potentials. For each particle, the operator represents the mean field created by all the others in the same state |φ〉.

Comment:

The Gross-Pitaevskii operator is simply a partial trace over the second particle:

(45)

where Pθ(2) is the projection operator Pθ(2) of the state of particle 2 onto |θ〉:

(46)

To show this, let us compute the partial trace on the right-hand side of (45). To obtain this trace (Complement E_III, § 5-b), we choose for particle 2 a set of basis states {|θ_n〉} whose first vector |θ₁〉 coincides with |θ〉:

(47)

Replacing Pθ(2) by its value (46) yields the product of δik (for the scalar product associated with particle 1) and δ_n1 (for the one associated with particle 2). This leads to:

(48)

which is simply the initial definition (40) of . Relation (45) is therefore another possible definition for the Gross-Pitaevskii potential.