Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

Quantum Mechanics, Volume 3

leads to the expression:

(97)

For the first scalar product to be non-zero, the subscript j must be such that νj = ν; in the same way, for the second product to be non-zero, we must have νj = ν′. For both conditions to be satisfied, we must impose ν = ν′, and the exchange term (93) is equal to:

(98)

where the summation is on all the values of j such that νj = ν: this term only exists if the two interacting particles are totally indistinguishable, which requires that they be in the same spin state (see the discussion in Complement E_XV).

We now define the direct and exchange potentials by:

(99)

The equalities (87) then lead to the Hartree-Fock equations in the position representation:

(100)

The general discussion of § 3-b can be applied here without any changes. These equations are both nonlinear and self-consistent, as the direct and exchange potentials are themselves functions of the solutions of the eigenvalue equations (100). This situation is reminiscent of the zero-temperature case, and we can, once again, look for solutions using iterative methods. The number of equations to be solved, however, is infinite and no longer equal to the finite number N, as already pointed out in § 3-c. The set of solutions must span the entire individual state space. Along the same line, in the definitions (99) of the direct and exchange potentials, the summations over j are not limited to N states, but go to infinity. However, even though the number of these wave functions is in principle infinite, it is limited in practice (for numerical calculations) to a high but finite number. As for the initial conditions to start the iteration process, one can choose for example the states and energies of a free fermion gas, but any other conjecture is equally possible.

Conclusion

There are many applications of the previous calculations, and more generally of the mean field theory. We give a few examples in the next complement, which are far from showing the richness of the possible application range. The main physical idea is to reduce, whenever possible, the calculation of the various physical quantities to a problem similar to that of an ideal gas, where the particles have independent dynamics. We have indeed shown that the individual level populations, as well as the total particle number, are given by the same distribution functions fβ as for an ideal gas – see relations (38) and (44). The same goes for the system entropy S, as already mentioned at the end of § 2-b-α. If we replace the free particle energies by the modified energies , the analogy with independent particles is quite strong.

If we now want to compute other thermodynamic quantities, as for example the average energy, we can no longer use the ideal gas formulas; we must go back to the equations of § 2-c. The grand potential may be calculated by inserting in (61) the |θi〉 and the obtained from the Hartree-Fock equations. Another method uses the fact that is given by ideal gas formulas that contain the distribution fβ, and hence do not require any further calculations. As:

(101)

we can integrate over μ (between –∞ and the current value μ, for a fixed value of μ) to obtain lnZ, and hence the grand potential. From this grand potential, all the other thermodynamic quantities can be calculated, taking the proper derivatives (for example a derivative with respect to β to get the average energy). We shall see an example of this method in § 4-a of the next complement.

We must however keep in mind that all these calculations derive from the mean field approximation, in which we replaced the exact equilibrium density operator by an operator of the form (32). In many cases this approximation is good, even excellent, as is the case, in particular, for a long-range interaction potential: each particle will interact with several others, therefore enhancing the averaging effect of the interaction potential. It remains, however, an approximation: if, for example, the particles interact via a “hard core” potential (infinite potential when the mutual distance becomes less than a certain microscopic distance), the particles, in the real world, can never be found at a distance from each other smaller than the hard core diameter; now this impossibility is not taken into account in (32). Consequently, there is no guarantee of the quality of a mean field approximation in all situations, and there are cases for which it is not sufficient.

1 ¹ They are not simply the juxtaposition of that complement’s equations: one could imagine writing those equations independently for each energy level, and then performing a thermal average. We are going to see (for example in § 2-d-β) that the determination of each level’s position already implies thermal averages, meaning that the levels are coupled.

2 ² Contrary to what is usually the case for a density operator, the trace of this reduced operator is not equal to 1, but to the average particle number — see relation (44). This different normalization is often more useful when studying systems composed of a large number of particles.

3 ³ We have changed the notation and of Chapter XV into and to avoid any confusion with the distribution functions fβ.

4 ⁴ For fermions, and when the temperature approaches zero, the distribution function included in the definition of ρI(1) becomes a step function and ρI(1) does indeed coincide with PN(1).

5 ⁵ The definition of partial traces is given in § 5-b of Complement EIII. The left hand side of (71) can be written as Σi, j 〈1 1: θi; 2: θj〉 dρI(1)O(1, 2) |1 : θi; 2 : θj〉. We then insert, after dρj(1), a closure relation on the kets |1 :θk;2 : θk′), with k′ = j since dρI(1) does not act on particle 2. This

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