Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 129

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).