Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 133

Assume the fermions we are studying are particles with spin 1/2, electrons for example. The basis {|r〉} of the individual states used in § 1 must be replaced by the basis formed with the kets {|r, ν)}, where ν is the spin index, which can take 2 distinct values noted ±1/2, or more simply ±. To the summation over d³r we must now add a summation over the 2 values of the index spin ν. A vector |φ〉 in the individual state space is now written:

(83)

with:

(84)

The variables r and ν play a similar role but the first one is continuous whereas the second is discrete. Writing them in the same parenthesis might hide this difference, and we often prefer noting the discrete index as a superscript of the function φ and write:

(85)

Let us build an N particle variational state from N orthonormal states , with n =1, 2,.. , N. Each of the describes an individual state including the spin and position variables; the first N₊ values of νn (n = 1,2,..N₊) are equal to +1/2, the last N_– are equal to –1/2, with N₊ + N_– = N (we assume N₊ and N_- are fixed for the moment but we may allow them to vary later to enlarge the variational family). In the space of the individual states, we introduce a complete basis whose first N kets are the , but where the subscript k varies from 1 to infinity⁷.

We assume the matrix elements of the external potential V₁ to be diagonal for ν; these two diagonal matrix elements can however take different values , which allows including the eventual presence of a magnetic field coupled with the spins. We also assume the particle interaction W₂ (1,2) to be independent of the spins, and diagonal in the position representation of the two particles, as is the case, for example, for the Coulomb interaction between electrons. With these assumptions, the Hamiltonian cannot couple states having different particle numbers N₊ and N_–.

Let us see what the general Hartree-Fock equations become in the {|r, ν〉} representation. In this representation, the effect of the kinetic and potential operators are well known. We just have to compute the effect of the Hartree-Fock potential WHF. To obtain its matrix elements, we use the basis to write the trace in (60):

(86)

As the right-hand side includes the scalar product which is equal to δkp, the sum over k disappears and we get:

(87)

(i) We first deal with the direct term contribution, hence ignoring in the bracket the term in P_ex(1, 2). We can replace the ket by its expression:

(88)

As the operator is diagonal in the position representation, we can write:

(89)

The direct term of (87) is then written:

(90)

where the scalar product of the bra and the ket is equal to . We finally obtain:

(91)

with:

(92)

This component of the mean field (Hartree term) contains a sum over all occupied states, whatever their spin is; it is spin independent.

(ii) We now turn to the exchange term, which contains the operator P_ex(1,2) in the bracket of (87). To deal with it, we can for example commute in (87) the two operators W₂(1, 2) and P_ex(1, 2); this last operator will then permute the two particles in the bra. Performing this operation in (90), we get, with the minus sign of the exchange term:

(93)

The scalar product will yield the products of δννp δνp ν′ δ(r – r₂), making the integral over d³r₂ disappear; this term is zero if ν ≠ ν′, hence the factor δνν′. Since W₂ (r′, r) = W₂(r, r′), we are left with:

(94)

where the sum is over the values of p for which νp = ν = ν′ (hence, limited to the first N₊ values of p, or the last N_–, depending on the case); the exchange potential has been defined as:

(95)

As is the case for the direct term, the exchange term does not act on the spin. There are however two differences. To begin with, the summation over p is limited to the states having the same spin v; second, it introduces a contribution which is non-diagonal in the positions (but without an integral), and which cannot be reduced to an ordinary potential (the term “non-local potential” is sometimes used to emphasize this property).

We have shown that the scalar product of equation (77) with 〈r, ν| introduces three potentials (in addition to the the one-body potential ), a direct potential V_dir(r) and two exchange potentials with ν = ±1/2. Equation (77) then becomes, in the {|r, ν〉} representation, a pair of equations:

(96)

These are the Hartree-Fock equations with spin and in the position representation, widely used in quantum physics and chemistry. It is not necessary to worry, in these equations, about the term in which the subscript p in the summation appearing in (92) and (95) is the same as the subscript n (of the wave function we are looking for); the contributions n = p cancel each other exactly in the direct and exchange potentials.

Both the “Hartree term” giving the direct potential contribution, and the “Fock term “ giving the exchange potential, can be interpreted in the same way as above (§ 1-f). The Hartree term contains the contributions of all the other electrons to the mean potential felt by one electron. The exchange potential, on the other hand, only involves electrons in the same spin state, and this can be simply interpreted: the exchange effect only occurs for two totally indistinguishable particles. Now if these particles are in orthogonal spin states, and as the interactions do not act on the spins, one can in principle determine which is which and the particles become distinguishable: the quantum exchange effects cancel out. As we already pointed out, the exchange potential is not a potential stricto sensu. It is not diagonal in the position representation, even though it basically comes from a particle interaction that is diagonal in position. It is the antisymmetrization of the fermions, together with the chosen variational approximation, which led to this peculiar non-diagonal form. It is however a Hermitian operator, as can be shown using the fact that the initial potential W₂(r, r′) is real and symmetric with respect to r and r′.