Set of supplementary notes.

136 CHAPTER 9. ELECTRONIC INSTABILITIES
spin configuration. There are complementary views of magnetism as originating either from
the alignment of local moments or from a spontaneous spin polarisation of itinerant electrons.
We begin with the former.
Direct exchange
As a first model system, let us consider two electrons in two orbitals, |a >, |b >, which
are orthogonal to each other, and which are eigenstates of the single-particle Hamiltonian Ĥ0.
Because the electrons are indistinguishable, the two-body wavefunction has to be antisymmetric
under particle exchange:
Ψ(r 1 , r 2 ) = −Ψ(r 2 , r 1 )
A simple approximation to the full two-body wavefunction can be formed from antisymmetrised
product wavefunctions:
Ψ(r 1 , r 2 ) = 1 √
2
(|ab〉 − |ba〉) ,
where the first slot in the Dirac-ket vector denotes the state occupied by electron 1, and the
second slot the state of electron 2.
If we now consider spin, as well, then we find four possible antisymmetrised two-particle
states, which can be grouped into one state with a singlet spin wavefunction, for which the
spatial state is symmetric under particle exchange
1
(|ab〉 + |ba〉)(| ↑↓〉 − | ↓↑〉)
2
and three states with triplet spin wavefunctions, for which the spatial state is antisymmetric
under particle exchange
⎛
1
2 (|ab〉 − |ba〉) ⎝
| ↑↑〉
| ↑↓〉 + | ↓↑〉
| ↓↓〉
We will now find that subject to the full Hamiltonian Ĥ = Ĥ0 + Ĥ1,2, where the interaction
part H 1,2 = V (r 1 − r 2 ), the singlet state has a higher energy than the triplet state.
We introduce some shorthand notation:
⎞
⎠
E 0 ≡ 〈ab|Ĥ|ab〉 = E a + E b + E Coul (9.15)
E Coul ≡ 〈ab|Ĥ1,2|ab〉
∫
(9.16)
= d 3 r 1 d 3 r 2 |ψ a (r 1 )| 2 |ψ b (r 2 )| 2 V (r 1 − r 2 ) (9.17)
E ex ≡ 〈ba|Ĥ1,2|ab〉
∫
(9.18)
= d 3 r 1 d 3 r 2 ψb ∗ (r 1 )ψ a (r 1 )ψa(r ∗ 2 )ψ b (r 2 )V (r 1 − r 2 ) (9.19)