Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 147

We now vary the ket |θk(t)〉 according to:

(30)

As in complement E_XV, we will only consider variations |δθk(t)〉 that lead to an actual variation of the ket ; those where |δθk(t)〉 is proportional to one of the occupied states |θl(t)〉 with l ≤ N yield no change for (or at the most to a phase change) and are thus irrelevant for the value of S. As we did in relations (32) or (69) of Complement E_XV, we assume that:

(31)

where δf(t) is an infinitesimal time-dependent function.

The computation is then almost identical to that of § 2-b in Complement E_XV. When |θk(t)〉 varies according to (31), all the other occupied states remaining constant, the only changes in the first line of (29) come from the terms i = k. In the second line, the changes come from either the i = k terms, or the j = k terms. As the W₂(1,2) operator is symmetric with respect to the two particles, these variations are the same and their sum cancels the 1/2 factor. All these variations involve terms containing either the ket eiχ |δθk(t)〉, or the bra 〈δθk(t)|e–iχ. Now their sum must be zero for any value of χ, and this is only possible if each of the terms is zero. Inserting the variation (31) of |θk(t)〉, and canceling the term in e–iχ leads to the following equality:

(32)

As we recognize in the function to be integrated the Hartree-Fock potential operator WHF(1, t) defined in (21), we can write:

(33)

with l > N.