Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji
alt="image"/>. This situation is reminiscent of the one encountered with the zero-temperature Hartree-Fock equations, and, a priori, no exact solutions can be found.
As for the zero-temperature case, we proceed by iteration: starting from a physically reasonable density operator
β. Validity limit
For a fermion system, there is no fundamental general limit for using the Hartree-Fock approximation. The pertinence of the final result obviously depends on the nature of the interactions, and whether a mean field treatment of these interactions is a good approximation. One can easily understand that the larger the interaction range, the more each particle will be submitted to the action of many others. This will lead to an averaging effect improving the mean field approximation. If, on the other hand, each particle only interacts with a single partner, strong binary correlations may appear, which cannot be correctly treated by a mean field acting on independent particles.
For bosons, the same general remarks apply, but the populations are no longer limited to 1 . When, for example, Bose-Einstein condensation occurs, one population becomes much larger than the others, and presents a singularity that is not accounted for in the calculations presented above. The Hartree-Fock approximation has therefore more severe limitations than for the fermions, and we now discuss this problem.
For a boson system in which many individual states have comparable populations, taking into account the interactions by the Hartree-Fock mean field yields as good an approximation as for a fermion system. If the system however is close to condensation, or already condensed, the mean field equations we have written are no longer valid. This is because the trial density operator in relation (31) contains a distribution function associated with each individual quantum state and varies as for an ideal gas, i.e. as an exponentially decreasing function of the occupation numbers. Now we saw in § 3-b-β of Complement BXV that, in an ideal gas, the fluctuations of the particle numbers in each of the individual states are as large as the average values of those particle numbers. If the individual state has a large population, these fluctuations can become very important, which is physically impossible in the presence of repulsive interactions. Any population fluctuation increases the average value of the square of the occupation number (equal to the sum of the squared average value and the squared fluctuation), and hence of the interaction energy (proportional to the average value of the square). A large fluctuation in the populations would lead to an important increase of the interaction repulsive energy, in contradiction with the minimization of the thermodynamic potential. In other words, the finite compressibility of the physical system, introduced by the interactions, prevents any large fluctuation in the density. Consequently, the fluctuations in the number of condensed particles predicted by the trial Hartree-Fock density operator are not physically acceptable, in the presence of condensation.
It is worth analyzing more precisely the origin of this Hartree-Fock approximation limit, in terms of correlations between the particles. Relation (51) concerns any two-particle operator
(89)
Its diagonal matrix elements are then written:
and are the sum of a direct term, and an exchange term. When i ≠ j, the presence of an exchange term is not surprising, and corresponds to the general discussion of § C-5 in Chapter XV. It is similar to the expression of the spatial correlation function written in (C-34) of that chapter, which is also the sum of two contributions, a direct one (C-32) and an exchange one (C-33). Since this last contribution is positive when r1 ≃ r2, the physical consequence of the exchange is a spatial bunching of the bosons. What is surprising though is that the exchange term still exists in (90) when i = j, even though the notion of exchange is meaningless: when dealing with a single individual state, the four expressions (C-21) of Chapter XV reduce to a single one, the direct term. We can also check that the exchange term (C-34) of Chapter XV includes the explicit condition i ≠ j, which means it receives no contribution from i = j. We shall furthermore confirm in § 3 of Complement AXVI that bosons all placed in the same individual quantum state are not spatially correlated, and therefore present neither bunching nor exchange effects. The mathematical expression of the trial two-particle Hartree-Fock density operator thus contains too many exchange terms. This does not really matter as long as the boson system remains far from Bose-Einstein condensation: the error involved is small since the x = j terms play a negligible role compared to the i ≠ j terms in the summations over i and j appearing in the interaction energy. However, as soon as an individual state becomes highly populated, significant errors can occur and the Hartree-Fock method must be abandoned. There exist, however, more elaborate theoretical treatments better adapted to this case.
3-c. Differences with the zero-temperature Hartree-Fock equations (fermions)
The main difference between the approach we just used and that of Complements CXV and EXV 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