Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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href="#ulink_d24a078f-2f16-5d24-8016-3b9a6e3bbac7">(40), induces a variation of :

(78)

      and thus leads to variations of expressions (61) of and . Their sum is:

      (79)

where the factor 1/2 in has been canceled since the variations induced by and double each other. Inserting (78) in this relation and using again (74), we get:

      (80)

      As for , its variation is the sum of a term in coming from the explicit presence of the energies in its definition (61), and a term in . If we let only the energy vary (not taking into account the variations of the distribution function), we get a zero result, since:

      (81)

      Consequently, we just have to vary by the distribution function, and we get:

      (82)

      Finally, after simplification by (which, by hypothesis, is different from zero), imposing the variation to be zero leads to the condition:

(83)

This expression does look like the stationarity condition at constant energy (77), but now the subscripts l and m are the same, and a term in is present in the operator.

3. Temperature dependent mean field equations

      Introducing a Hartree-Fock operator acting in the single particle state space allows writing the stationarity relations just obtained in a more concise and manageable form, as we now show.

3-a. Form of the equations

      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

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