Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 164

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 .