Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 162

We now consider an arbitrary two-particle operator Ĝ and compute its average value with the density operator . The general expression of a symmetric two-particle operator is given by relation (C-16) of Chapter XV, and we can write:

(45)

We follow the same steps as in § 2-a of Complement E_XV: we use the mean field approximation to replace the computation of the average value of a two-particle operator by that of average values for one-particle operators. We can, for example, use relation (43) of Complement B_XV, which shows that:

(46)

We then get:

(47)

Which, according to (40), can also be written as:

(48)

where P_ex is the exchange operator between particles 1 and 2. Since:

(49)

and as the operators (1) and (2) are diagonal in the basis |θi〉, we can write the right-hand side of (48) as:

(50)

which is simply a (double) trace on two particles 1 and 2. This leads to:

(51)

As announced above, the average value of the two-particle operator Ĝ can be expressed, within the Hartree-Fock approximation, in terms of the one-particle reduced density operator (1); this relation is not linear.

Comment:

The analogy with the computations of Complement Exv becomes obvious if we regroup its equations (57) and (58) and write:

(52)

Replacing W₂(1, 2) by G, we get a relation very similar to (51), except for the fact that the projectors PN must be replaced by the one-particle operators . In § 3-d, we shall come back to the correspondence between the zero and non-zero temperature results.