Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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in the trace only act on that same particle, numbered arbitrarily 1; the subscript 1 could obviously be replaced by the subscript of any other particle, since they all play the same role. The average potential energy coming from the external potential is computed in a similar way and can be written as:

(53)

       β. Average interaction energy, Hartree-Fock potential operator

      The average interaction energy can be computed using the general expression (C-16) of Chapter XV for any two-particle operator, which yields:

      (54)

      For the average value in the Fock state to be different from zero, the operator must leave unchanged the populations of the individual states |θn〉 and |θq〉. As in § C-5-b of Chapter XV, two possibilities may occur: either r = n and s = q (the direct term), or r = q and s = n (the exchange term). Commuting some of the operators, we can write:

      (55)

where nr and ns are the respective populations of the states |θr〉 and |θs〉. Now these populations are different from zero only if the subscripts r and s are between 1 and N, in which case they are equal to 1 (note also that we must have r ≠ s to avoid a zero result). We finally get⁵:

      (56)

      (the constraint i ≠ j may be ignored since the right-hand side is equal to zero in this case). Here again, the subscripts 1 and 2 label two arbitrary, but different particles, that could have been labeled arbitrarily. We can therefore write:

(57)

where P_ex(1,2) is the exchange operator between particle 1 and 2 (the transposition which permutes them). This result can be written in a way similar to (53) by introducing a “Hartree-Fock potential” WHF, similar to an external potential acting in the space of particle 1; this potential is defined as the operator having the matrix elements:

(58)

      This operator is Hermitian, since, as the two operators P_ex and W₂ are Hermitian and commute, we can write:

      (59)

Furthermore, we recognize in (58) the matrix element of a partial trace on particle 2 (Complement E_III, § 5-b):

(60)

where the projector PN has been introduced inside the trace to limit the sum over j to its first N terms, as in (57). The one-particle operator WHF(1) is thus the partial trace over a second particle (with the arbitrary label 2) of a product of operators acting on both particles. As the summation over j is now taken into account, we are left in (57) with a summation over i, which introduces a trace over the remaining particle 1, and we get:

(61)

This average value depends on the subspace chosen with the variational ket in two ways: explicitly as above, via the projector PN(1) that shows up in the average value (61), but also implicitly via the definition of the Hartree-Fock potential in (60).

       ϒ. Role of the one-particle reduced density operator

All the average values can be expressed in terms of the projector PN onto the subspace of the space the individual states spanned by the N individual states |θ₁〉, |θ₂〉, ….|θN〉, which means, according to (50), in terms of the one-particle reduced density operator . Hence it is this operator that is the pertinent variable to optimize rather than the set of individual states: certain variations of those states do not change PN, and are meaningless for our purpose.

Furthermore, the choice of the trial ket is equivalent to that of PN. In other words, the variational ket built in (1) does not depend on the basis chosen in the subspace : if we choose in this subspace any orthonormal basis {|uj〉} other than the {|θj〉} basis, and if we replace in (1) the by the , the ket will remain the same (to within a non-relevant phase factor) as we now show. As seen in § A-6 of Chapter XV, each operator is a linear combination of the , so that in the product of all the

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