Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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with the kinetic energy contribution Ĥ₀ in (1). We call K₀ the individual kinetic energy operator:

      (55)

      (m is the particle mass). Equality (43) applied to Ĥ₀ yields the average kinetic energy when the system is described by :

(56)

      This result is easily interpreted; each individual state contributes its average kinetic energy, multiplied by its population.

      The computation of the average value follows the same steps:

(57)

      (as in Complement E_XV, operator V₁ is the one-particle external potential operator).

      To complete the calculation of the average value of Ĥ, we now have to compute the trace , the average value of the interaction energy when the system is described by . Using relation (51) we can write this average value as a double trace:

(58)

We now turn to the average value of . The calculation is simplified since is, like Ĥ₀, a one-particle operator; furthermore, the |θi〉 have been chosen to be the eigenvectors of with eigenvalues – see relation (26). We just replace in (56), Ĥ₀ by , and obtain:

      (59)

Regrouping all these results and using relation (36), we can write the variational grand potential as the sum of three terms:

      (60)

      with:

(61)

2-d. Optimization

      We now vary the eigenenergies and eigenstates |θk〉 of to find the value of the density operator that minimizes the average value of the potential. We start with the variations of the eigenstates, which induce no variation of . The computation is actually very similar to that of Complement E_XV, with the same steps: variation of the eigenvectors, followed by the demonstration that the stationarity condition is equivalent to a series of eigenvalue equations for a Hartree-Fock operator (a one-particle operator). Nevertheless, we will carry out this computation in detail, as there are some differences. In particular, and contrary to what happened in Complement E_XV, the number of states |θi〉 to be varied is no longer fixed by the particle number N; these states form a complete basis of the individual state space, and their number can go to infinity. This means that we can no longer give to one (or several) state(s) a variation orthogonal to all the other |θj〉; this variation will necessarily be a linear combination of these states. In a second step, we shall vary the energies .

       α. Variations of the eigenstates

      As the eigenstates |θi〉 vary, they must still obey the orthogonality relations:

      (62)

      The simplest idea would be to vary only one of them, |θl〉 for example, and make the change:

      (63)

      The orthogonality conditions would then require:

      (64)

preventing |dθl〉 from having a component on any ket |θi〉 other than |θl〉: in other words, |dθl〉 and |θl〉 would be colinear. As |θl〉 must remain normalized, the only possible variation would thus be a phase change, which does not affect either the density operator (1) or any average values computed with . This variation does not change anything and is therefore irrelevant.

      It is actually more interesting to vary simultaneously two eigenvectors, which will be called |θl〉 and |θm〉, as it is now possible to give |θl〉 a component on |θm〉, and the reverse. This does not change the two-dimensional subspace spanned by these two states; hence their orthogonality with all the other basis vectors is automatically preserved. Let us give the two vectors the following infinitesimal variations (without changing their energies and ):

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