Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

³:

(32)

      In this case, for the second matrix element to be non-zero, both subscripts m and n must be equal to 1 and the same is true for both subscripts k and l (otherwise the operator will yield a Fock state orthogonal to ). When all the subscripts are equal to 1, the operator multiplies the ket by N (N — 1). This leads to:

(33)

      The average interaction energy is therefore simply the product of the number of pairs N(N —1)/2 that can be formed with N particles and the average interaction energy of a given pair.

We can replace |θ₁〉 by |θ〉, since they are equal. The variational energy, obtained as the sum of (30), (31) and (33), then reads:

(34)

3-b. Energy minimization

      Consider a variation of |θ〉:

(35)

      where |δα〉 is an arbitrary infinitesimal ket of the individual state space, and χ an arbitrary real number. To ensure that the normalization condition (6) is still satisfied, we impose |δα〉 and |θ〉 to be orthogonal:

(36)

so that 〈θ|θ〉 remains equal to 1 (to the first order in |δα〉). Inserting (35) into (34) to obtain the variation of the variational energy, we get the sum of two terms: the first one comes from the variation of the ket |θ〉 and is proportional to eiχ the second one comes from the variation of the bra 〈θ| and is proportional to e–χ. The result has the form:

      (37)

      The stationarity condition for must hold for any arbitrary real value of χ. As before (§ 2-b-α), it follows that both δc₁ and δc₂ are zero. Consequently, we can impose the variation to be zero as just the bra 〈ι| varies (but not the ket |θ〉), or the opposite.

      Varying only the bra, we get the condition:

      (38)

As the interaction operator W₂ (1, 2) is symmetric, the last two terms within the bracket in this equation are equal. We get (after simplification by N):

(39)

3-c. Gross-Pitaevskii equation

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