Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 173

Let us check that the Hartree-Fock method for non-zero temperature yields the same results as the zero temperature method explained in Complement E_XV for fermions.

In § 2-d of Complement B_XV, we introduced for an ideal gas the concept of a degenerate quantum gas. It can be generalized to a gas with interactions: in a fermion system, when βμ ≫ 1, the system is said to be strongly degenerate. As the temperature goes to zero, a fermion system becomes more and more degenerate. Can we be certain that the results of this complement are in agreement with those of Complement E_XV, valid at zero temperature?

We saw that the temperature comes into play in the definition (85) of the mean Hartree-Fock potential, WHF. In the limit of a very strong degeneracy, the Fermi-Dirac distribution function appearing in the definition (40) of becomes practically a step function, equal to 1 for energies ej less than the chemical potential μ, and zero otherwise (Figure 1 of Complement B_XV. In other words, the only populated states (and by a single fermion) are the states having energies less than μ, i.e. less than the Fermi level. Under such conditions, the of (84) becomes practically equal to the projector PN(2) which, in Complement E_XV, appears in the definition (52) of the zero-temperature Hartree-Fock potential; in other words, the partial trace appearing in this relation (85) is then strictly limited to the individual states having the lowest energies. We thus obtain the same Hartree Fock equations as for zero temperature, leading to the determination of a set of individual eigenstates on which we can build a unique N-particle state.