Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

Quantum Mechanics, Volume 3

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)

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