Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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(8)

      An assembly of bosons that occupy the same individual state is called a “Bose-Einstein condensate”.

Relation (8) shows that the kets are normalized to 1. We are going to vary |θ〉, and therefore , so as to minimize the average energy:

      (9)

2. First approach

      We start with a simple case where the bosons have no spin. We can then use the wave function formalism and keep the computations fairly simple.

2-a. Trial wave function for spinless bosons, average energy

      Assuming one single individual state to be populated, the wave function Ψ(r₁, r₂,…, r_N) is simply the product of N functions θ(r):

      (10)

      with:

      (11)

      This wave function is obviously symmetric with respect to the exchange of all particles and can be used for a system of identical bosons.

In the position representation, each operator K₀(q) defined by (3) corresponds to (—ħ²/2m) Δ_rq , where Δ_rq is the Laplacian with respect to the position r_q; consequently, we have:

      (12)

      In this expression, all the integral variables others than r_q simply introduce the square of the norm of the function θ(r), which is equal to 1. We are just left with one integral over r_q, in which r_q plays the role of a dummy variable, and thus yields a result independent of q. Consequently, all the q values give the same contribution, and we can write:

(13)

      As for the one-body potential energy, a similar calculation yields:

(14)

      Finally, the interaction energy calculation follows the same steps, but we must keep two integral variables instead of one. The final result is proportional to the number N(N — 1)/2 of pairs of integral variables:

(15)

      The variational average energy is the sum of these three terms:

(16)

2-b. Variational optimization

      We now optimize the energy we just computed, so as to determine the wave functions θ(r) corresponding to its minimum value.

       α. Variation of the wave function

      Let us vary the function θ(r) by a quantity:

      (17)

      where δf(r) is an infinitesimal function and χ an arbitrary number. A priori, δf(r) must be chosen to take into account the normalization constraint (6), which forces the integral of the θ(r) modulus squared to remain constant. We can, however, use the Lagrange multiplier method (Appendix V) to impose this constraint. We therefore introduce the multiplier μ (we shall see in § 4-a that this factor can be interpreted as the chemical potential) and minimize the function:

(18)

This allows considering the infinitesimal variation δf(r) to be free of any constraint. The variation of the function is now the sum of 4 variations, coming from the three terms of (16) and from the integral in (18). For example, the variation of yields:

      (19)

      which is the sum of a term proportional to e^–iχ and another proportional to e^iχ. This is true for all 4 variations and the total variation can be expressed as the sum of two terms:

      (20)

the first being the δf^*(r) contribution and the second, that of δf(r). Now if is stationary, must be zero whatever the choice of χ, which is real. Choosing for example χ = 0 imposes δc₁ + δc₂ = 0, and the choice χ = π/2 leads

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