(32) has the same form as relation (5) of Complement BXV, with a simple change: the replacement of the free particle energies ek = ħ2k2/2m by the energies , which are as yet unknown. As this change does not impact the mathematical structure of the density operator, we can directly use the results of Complement BXV.
α. Variational partition function
The function only depends on the variational energies , since the trace of (32) may be computed in the basis {}, which yields:
(33)
We simply get an expression similar to relation (7) of Complement BXV, obtained for an ideal gas. Since for fermions nk can only take the values 0 and 1, we get:
(34)
whereas for bosons nk varies from 0 to infinity, so that:
Computing the entropy can be done in a similar way. As the density operator has the same form as the one describing the thermal equilibrium of an ideal gas, we can use for a system described by the formulas obtained for the entropy of a system without interactions.
β. One particle, reduced density operator
Let us compute the average value of with the density operator :
where the distribution function fβ is noted for fermions, and for bosons:
(39)
When the system is described by the density operator the average populations of the individual states are therefore determined by the usual Fermi-Dirac or Bose-Einstein distributions. From now on, and to simplify the notation, we shall write simply |θk〉 for the kets .
We can introduce a “one-particle reduced density operator” (1) by2:
where the 1 enclosed in parentheses and the subscript 1 on the left-hand side emphasize we are dealing with an operator acting in the one-particle state space (as opposed to that acts in the Fock space); needless to say, this subscript has nothing to do with the initial numbering of the particles, but simply refers to any single particle among all the system particles. The diagonal elements of (1) are the individual state populations. With this operator, we can compute the average value over of any one-particle operator :
(41)
as we now show. Using the expression (B-12) of Chapter XV for any one-particle operator3, as well as (38), we can write:
As we shall see, the density operator (1) is quite useful since it allows obtaining in a simple way all the average values that come into play in the Hartree-Fock computations. Our variational calculations will simply amount to varying (1). This operator presents, in a certain sense, all the properties of the variational density operator chosen in (28) in the Fock space. It plays the same role4
(32) has the same form as relation (5) of Complement BXV, with a simple change: the replacement of the free particle energies ek = ħ2k2/2m by the energies 