2.5 Change of the One-Particle Representation
29
ˆ
ψ
†
(ξ) =
q
c
†
q ψ
∗
q (ξ)
ˆ
ψ(ξ) =
q
c q ψ q (ξ)
(2.45)
creating or destructing, respectively, a particle with spin σ at the position x. These
operators, also referred to as field operators, obey the commutation relations
{ ˆ
ψ
†
(ξ), ˆ
ψ(ξ
)} = δ(x − x
)δ σσ
{ ˆ
ψ(ξ), ˆ
ψ(ξ
)} = 0, { ˆ
ψ
†
(ξ), ˆ
ψ
†
(ξ
)} = 0
(2.46)
In terms of field operators, the one- and two-particle operators take on the forms
ˆ
W =
dξ ˆ
w(ξ) ˆ
ψ
†
(ξ) ˆ
ψ(ξ)
ˆ
V =
1
2
dξ dξ
ˆ
v(ξ, ξ
) ˆ
ψ
†
(ξ) ˆ
ψ
†
(ξ
) ˆ
ψ(ξ
) ˆ
ψ(ξ)
(2.47)
and the N -electron coordinate eigenstate (1.21) can be written as
|ξ 1 . . . ξ N = ˆ
ψ
†
(ξ N ) . . . ˆ
ψ
†
(ξ 1 )|∅
(2.48)
A useful mixed representation, associated with the products |xχ γ (σ) of position
operator eigenstates and spin-functions, is given according to
ˆ
ψ
†
γ (x) =
q
c
†
qγ ϕ
∗
q (x), ˆ
ψ γ (x) =
q
c qγ ϕ q (x)
(2.49)
Here, the original spin-orbital quantum number q has been expanded into the pair
of a spatial and a spin quantum number, q → qγ, and ϕ q (x) is the spatial orbital in
the spin-orbital ψ qγ (ξ) as in Eq. (1.61).
Exercises
2.1 Verify the SC rules c1–c4 for the two-particle operator (2.20) using the approach
discussed in Sect. 2.4.
2.2 (a) Evaluate the matrix element
N −1
j
| ˆ
H |
N −1
akl for the cases k = j and l = j.
Show that for HF orbitals the result is as given by Eq. (2.37).
(b) Evaluate the matrix elements
N −1
j
| ˆ
V |
N −1
abklm and aj | ˆ
V | bcdklm .
2.3 (a) Evaluate the commutator [c j , ˆ
V ] where ˆ
V is a two-particle operator in the
form (2.20).
(b) Evaluate the commutator [ ˆ
A, ˆ
B] for the one-particle operators
ˆ
A =
r,s a rs c
†
r c s , ˆ
B =
u,v b uv c
†
u c v .