Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 81

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)