Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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that is zero. As for , an integration by parts over time shows that it is the complex conjugate of , and therefore also equal to zero. The functional S is thus stationary in the vicinity of any exact solution of the Schrödinger equation.

      Suppose we choose any variational family of normalized kets , but which now includes a ket for which S is stationary. A simple example is the case where is a family that contains the exact solution of the Schrödinger equation; according to what we just saw, this exact solution will make S stationary, and conversely, the ket that makes S stationary is necessarily . In this case, imposing the variation of S to be zero allows identifying, inside the family , the exact solution we are looking for. If we now change the family continuously from to , in general will no longer contain the exact solution of the Schrödinger equation. We can however follow the modifications at all times of the values of the ket . Starting from an exact solution of the equation, this ket progressively changes, but, by continuity, will stay in the vicinity of this exact solution if stays close to . This is why annulling the variation of S in the family is a way of identifying a member of that family whose evolution remains close to that of a solution of the Schrödinger equation. This is the method we will follow, using the Fock states as a particular variational family.

2-c. Particular case of a time-independent Hamiltonian

      If the Hamiltonian H is time-independent, one can look for time-independent kets to make the functional S stationary. The function to be integrated in the definition of the functional S also becomes time-independent, and we can write S as:

      (17)

      Since the two times t₀ and t₁ are fixed, the stationarity of S is equivalent to that of the diagonal matrix element of the Hamiltonian . We find again the stationarity condition of the time-independent variational method (Complement E_XI), which appears as a particular case of the more general method of the time-dependent variations. Consequently, it is not surprising that the Hartree-Fock methods, time-dependent or not, lead to the same Hartree-Fock potential, as we now show.

3. Computing the optimizer

The family of the state vectors we consider is the set of Fock kets defined in (1). We first compute the function to be integrated in the functional (10) when |Ψ(t)〉 takes the value .

3-a. Average energy

For the term in H(t), the calculation is identical to the one we already did in § 1-b of Complement E_XV. We first add to the series of orthonormal states |θi (t)〉 with i = 1, 2, …, N other orthonormal states |θi (t)〉 with i = N + 1, N + 2, …, to obtain a complete orthonormal basis in the space of individual states. Using this basis, we can express the one-particle and two-particle operators according to relations (B-12) and (C-16) of Chapter XV. This presents no difficulty since the average values of creation and annihilation operator products are easily obtained in a Fock state (they only differ from zero if the product of operators leaves the populations of the individual states unchanged). Relations (52), (53) and (57) of Complement E_XV are still valid when the |θi〉 become time-dependent. We thus get for the average kinetic energy:

      (18)

      for the external potential energy:

      (19)

      and for the interaction energy:

(20)

3-b. Hartree-Fock potential

We recognize in (20) the diagonal element (i = k) of the Hartree-Fock potential operator WHF(1, t) whose matrix elements have been defined in a general way by relation (58) of Complement E_XV:

(21)

      We also noted in that complement E_XV that WHF (1, t) is a Hermitian operator.

      It is often handy to express the Hartree-Fock potential using a partial trace:

      (22)

      where PN is the projector onto the subspace spanned by the N kets |θi(t)〉:

      (23)

      As we have seen before, this projector is actually nothing bu the one-particle reduced density operator

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