Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 168

Let us define a temperature dependent Hartree-Fock operator as the partial trace that appears in the previous equations:

(84)

It is thus an operator acting on the single particle 1. It can be defined just as well by its matrix elements between the individual states:

(85)

Equation (77) is valid for any two chosen values l and m, as long as . When l is fixed and m varies, it simply means that the ket:

(86)

is orthogonal to all the eigenvectors |θm〉 having an eigenvalue different from ; it has a zero component on each of these vectors. As for equation (83), it yields the component of this ket on |θl〉, which is equal to . The set of |θm〉 (including those having the same eigenvalue as |θl〉) form a basis of the individual state space, defined by (26) as the basis of eigenvectors of the individual operator . Two cases must be distinguished:

(i) If is a non-degenerate eigenvalue of , the set of equations (77) and (83) determine all the components of the ket [K0 + V₁ + WHF(β)]|θl〉). This shows that |θl〉 is an eigenvector of the operator K₀ + V₁ + WHF with the eigenvalue .

(ii) If this eigenvalue of is degenerate, relation (77) only proves that the eigen-subspace of , with eigenvalue , is stable under the action of the operator K₀ + V₁ + WHF it does not yield any information on the components of the ket (86) inside that subspace. It is possible though to diagonalize K₀ + V₁ + WHF inside each of the eigen-subspace of , which leads to a new eigenvectors basis |φn〉, now common to and K₀ + V₁ + WHF.

We now reason in this new basis where all the [K₀ + V₁ + WHF(β)]|φn〉 are proportional to |φn〉. Taking (83) into account, we get:

(87)

As we just saw, the basis change from the |θl〉 to the |φn〉 only occurs within the eigen-subspaces of corresponding to given eigenvalues ; one can then replace the |θl〉 by the |φn〉 in the definition (40) of and write:

(88)

Inserting this relation in the definition (84) of WHF(β) leads to a set of equations only involving the eigenvectors |φn〉.

For all the values of n we get a set of equations (87), which, associated with (84) and (88) defining the potential WHF(β) as a function of the |φn〉, are called the temperature dependent Hartree-Fock equations.