Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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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 + 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.

3-b. Properties and limits of the equations

      We now discuss how to apply the mean field equations we have obtained, and their limit of validity, which are more stringent for bosons than for fermions.

       α. Using the equations

      Hartree-Fock equations concern a self-consistent and nonlinear system: the eigenvectors |φn〉 and eigenvalues of the density operator are solutions of an eigenvalue equation (87) which itself depends on Скачать книгу