Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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.

2-c. Variational grand potential

We now have to compute the grand potential written in (30). As the exponential form (28) for the trial operator makes it easy to compute ln, we see that the terms in μ cancel out, and we get:

      (53)

We now have to compute the average energy, with the density operator , of the difference between the Hamiltonians Ĥ and respectively defined by (1) and (25).

      We first compute the trace:

      (54)

      starting with the kinetic energy contribution Ĥ₀ in (1). We call K₀ the individual kinetic energy operator:

      (55)

      (m is the particle mass). Equality (43) applied to Ĥ₀ yields the average kinetic energy when the system is described by :

(56)

      This result is easily interpreted; each individual state contributes its average kinetic energy, multiplied by its population.

      The computation of the average value follows the same steps:

(57)

      (as in Complement E_XV, operator V₁ is the one-particle external potential operator).

      To complete the calculation of the average value of Ĥ, we now have to compute the trace , the average value of the interaction energy when the system is described by . Using relation (51) we can write this average value as a double trace:

(58)

We now turn to the average value of . The calculation is simplified since is, like Ĥ₀,