Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 121

The average value of the interaction energy in the state has already been computed in § C-5 of Chapter XV. We just have to replace, in the relations (C-28) as well as (C-32) to (C-34) of that chapter, the ni by 1 for all the occupied states |θi〉, by zero for the others, and to rename the wave functions ui(r) as θi(r). We then get:

(18)

We have left out the condition i ≠ j no longer useful since the i = j terms are zero. The second line of this equation contains the sum of the direct and the exchange terms.

The result can be written in a more concise way by introducing the projector PN over the subspace spanned by the N kets |θi〉:

(19)

Its matrix elements are:

(20)

This leads to:

(21)

Comment:

The matrix elements of PN are actually equal to the spatial non-diagonal correlation function G₁(r, r′), which will be defined in Chapter XVI (§ B-3-a). This correlation function can be expressed as the average value of the product of field operators Ψ(r):

(22)

For a system of N fermions in the states |θ₁〉, |θ₂〉, ..,|θN〉, we can write:

(23)

Inserting this relation in (18) we get:

(24)

Comparison with relation (C-28) of Chapter XV, which gives the same average value, shows that the right-hand side bracket contains the two-particle correlation function G₂(r, r′). For a Fock state, this function can therefore be simply expressed as two products of one-particle correlation functions at two points:

(25)