Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 123

Operator can be diagonalized in the subspace , as can be shown³ from its definition (36) – a more direct demonstration will be given in § 2. We call φn(r) its eigenfunctions. These functions φn(r) are linear combinations of the θj(r) corresponding to the states appearing in the trial ket (1), and therefore lead to the same N-particle state, because of the antisymmetrization⁴. The basis change from the θi(r) to the φn(r) has no effect on the projector PN onto to the subspace , whose matrix elements appearing in (36) can be expressed in a way similar to those in (20):

(37)

Consequently, the eigenfunctions of the operator obey the equations:

(38)

where are the associated eigenvalues. These relations are called the “Hartree-Fock equations”.

For the average total energy associated with a state such as (1) to be stationary, it is therefore necessary for this state to be built from N individual states whose orthogonal wave functions φ₁, φ₂, .. , φN are solutions of the Hartree-Fock equations (38) with n = 1, 2, .. , N. Conversely, this condition is sufficient since, replacing the θj(r) by solutions φn(r) of the Hartree-Fock equations in the energy variation (34) yields the result:

(39)

which is zero for all δφk(r) variations, since, according to (32), they must be orthogonal to the N solutions φn(r). Conditions (38) are thus equivalent to energy stationarity.