Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

Quantum Mechanics, Volume 3

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.

3-d. Zero-temperature limit (fermions)

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.

3-e. Wave function equations

Let us write the Hartree-Fock equations (87) in terms of wave functions: these equations are strictly equivalent to (87), written in terms of operators and kets, but their form is sometimes easier to use, in particular for numerical calculations.

Assuming the particles have a spin, we shall note the wave functions φν(r), with:

(91)

where the spin quantum number ν can take (2S + 1) values; according to the nature of the particles, the possible spins S are S = 0, S = 1/2, S = 1 etc. As in Complement E_XV (§ 2-d), we introduce a complete basis for the individual state space, built from kets that are all eigenvectors of the spin component along the quantization axis, with eigenvalue νk. For each value of n, the spin index ν takes on a given value νn and is not, therefore, an independent index. As for the potentials, we assume here again that V₁ is diagonal in ν, but that its diagonal elements may depend on ν. The interaction potential, however, is described by a function W₂ (r, r′) that only depends on r – r′, but does not act on the spins.

To obtain the matrix elements of in the representation {|r, ν)}, we use (85) after replacing the |θ〉 by the |φ〉 (we showed in § 3 that this was possible). We now multiply both sides by and , and sum over the subscripts k and l; we recognize in both sides the closure relations:

(92)

This leads to:

(93)

As in § C-5 of Chapter XV, we get the sum of a direct term (the term 1 in the central bracket) and an exchange term (the term in ηP_ex). This expression contains the same matrix element as relation (87) of Complement E_xv, the only difference being the presence of a coefficient in each term of the sum (plus the fact that the summation index goes to infinity).

(i) For the direct term, as we did in that complement, we insert a closure relation on the particle 2 position:

(94)

Since the interaction operator is diagonal in the position representation, the part of the matrix element of (93) that does not contain the exchange operator becomes:

(95)

The direct term of (93) is then written:

(96)

which is equivalent to relation (91) of Complement E_XV.

(ii) The exchange term is obtained by permutation of the two particles in the ket appearing on the right-hand side of (93); the diagonal

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