Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 172

The main difference between the approach we just used and that of Complements C_XV and E_XV is that these complements were only looking for a single eigenstate of the Hamiltonian Ĥ, generally its ground state. If we are now interested in several of these states, we have to redo the computation separately for each of them. To study the properties of thermal equilibrium, one could imagine doing the calculations a great many times, and then weigh the results with occupation probabilities. This method obviously leads to heavy computations, which become impossible for a macroscopic system having an extremely large number of levels. In the present complement, the Hartree-Fock equations yield immediately thermal averages, as well as eigenvectors of a one-particle density operator with their energies.

Another important difference is that the Hartree-Fock operator now depends on the temperature, because of the presence in (85) of a temperature dependent distribution function – or, which amounts to the same thing, of the presence in (84) of an operator dependent on β, and which replaces the projector PN (2) onto all the populated individual states. The equations obtained remind us of those governing independent particles, each finding its thermodynamic equilibrium while moving in the self-consistent mean field created by all the others, also including the exchange contribution (which can be ignored in the simplified “Hartree” version).

We must keep in mind, however, that the Hartree-Fock potential associated with each individual state now depends on the populations of an infinity of other individual states, and these populations are function of their energy as well as of the temperature. In other words, because of the nonlinear character of the Hartree-Fock equations, the computation is not merely a juxtaposition of separate mean field calculations for stationary individual states.