Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji
3-c. Time derivative
As for the time derivative term, the function it contains can be written as:
(26)
In this summation, all terms involving the individual states j other than the state i (which is undergoing the derivation) lead to an expression of the type:
(27)
which equals 1 since this expression is the square of the norm of the state
(28)
3-d. Functional value
Regrouping all these results, we can write the value of the functional S in the form:
4. Equations of motion
We now vary the ket |θk(t)〉 according to:
(30)
As in complement EXV, we will only consider variations |δθk(t)〉 that lead to an actual variation of the ket
where δf(t) is an infinitesimal time-dependent function.
The computation is then almost identical to that of § 2-b in Complement EXV. 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:
As we recognize in the function to be integrated the Hartree-Fock potential operator WHF(1, t) defined in (21), we can write:
with l > N.
4-a. Time-dependent Hartree-Fock equations
As the choice of the function δf(t) is arbitrary, for expression (33) to be zero for any δf(t) requires the function inside the curly brackets to be zero at all times t. Stationarity therefore requires the ket:
(34)
to have no components on any of the non-occupied states |θl(t)〉 with (l > N). In other words, stationarity will be obtained if, for all values of k & between 1 and N, we have:
(35)
where |ξk(t)〉 is any linear combination of the occupied states |θl(t)〉 (l < N). As we pointed out at the beginning of § 4, adding to one of the |θk(t)〉 a component on the already occupied individual states has no effect on the Af-particle state (aside from an eventual change of phase), and therefore does not change the value of S; consequently, the stationarity of this functional does not depend on the value of the ket |ξk(t)〉, which can be any ket, for example the zero ket.
Finally, if the |θn(t)〉 are equal to the solutions |φn(t)〉 of the N equations:
the functional S is indeed stationary for all times. Furthermore, as we saw in Complement Exv that WHF(t) is Hermitian, so is the operator on the right-hand side of (36). Consequently, the N kets |φn(t)〉 follow an evolution similar to the usual Schrödinger evolution, described by a unitary evolution operator (Complement FIII). Such an operator does not change either the norm nor the scalar products of the kets: if the kets |φn(t)〉x initially formed an orthonormal set, this remains true at any later time. The whole calculation just presented is thus consistent; in particular, the norm of the N-particle state vector
Relations (36) are the time-dependent Hartree-Fock equations. Introducing the one-particle mean field operator allowed us not only to compute the stationary energy levels, but also to treat time-dependent problems.
4-b. Particles in a single spin state
Let us return to the particular case of fermions all having the same spin state, as in § 1 of Complement EXV. We can then write the Hartree-Fock equations in terms of the wave functions as: