22
2 Second Quantization
(c p c
†
q + c
†
q c p )|q 1 . . . q N = δ pq |q 1 . . . q N
Since this holds for arbitrary basis states, we may conclude the operator identity
c
†
q , c p
= c p c
†
q + c
†
q c p = δ pq
(2.12)
In a similar way, the relations
c
†
p , c
†
q
= 0,
c p , c q
= 0
(2.13)
can be derived. The anticommutator relations (2.12) and (2.13) establish a versatile
tool for handling the many-electron product states (2.5) in an algebraic manner.
2.3 Operators in Second Quantization
The creation and destruction operators of second quantization allow us to represent
physical operators in a very advantageous way, as will be discussed in the following.
Let us first consider operators of the one-particle type (Eq. 1.32):
ˆ
W =
N
i=1
ˆ
w(i)
(2.14)
In second quantization, the operator ˆ
W can be written in a more general form, which
is no longer referring to a specific N -electron space:
ˆ
W
=
p,q
w pq c
†
p c q
(2.15)
where w pq = =p| ˆ
w|q denote the one-particle integrals (1.46). To prove the equivalence of the two forms (for a specific electron number N ), let us inspect the matrix
elements of ˆ
W
with respect to the basis set of N -electron product states. Using the
anticommutator algebra established above, the result of applying ˆ
W
to a general
product state becomes
p,q
w pq c
†
p c q |q 1 . . . q N =
p
q i
w pq i c
†
p c q i |q 1 . . . q N
=
p
q i
w pq i |q 1 . . . q i−1 pq i+1 . . . q N
Note that here no overall sign change occurs because a pair of operators is moved
to the respective position in the product state. Since, according to the SC rules a1
and a2,
2 Second Quantization
(c p c
†
q + c
†
q c p )|q 1 . . . q N = δ pq |q 1 . . . q N
Since this holds for arbitrary basis states, we may conclude the operator identity
c
†
q , c p
= c p c
†
q + c
†
q c p = δ pq
(2.12)
In a similar way, the relations
c
†
p , c
†
q
= 0,
c p , c q
= 0
(2.13)
can be derived. The anticommutator relations (2.12) and (2.13) establish a versatile
tool for handling the many-electron product states (2.5) in an algebraic manner.
2.3 Operators in Second Quantization
The creation and destruction operators of second quantization allow us to represent
physical operators in a very advantageous way, as will be discussed in the following.
Let us first consider operators of the one-particle type (Eq. 1.32):
ˆ
W =
N
i=1
ˆ
w(i)
(2.14)
In second quantization, the operator ˆ
W can be written in a more general form, which
is no longer referring to a specific N -electron space:
ˆ
W
=
p,q
w pq c
†
p c q
(2.15)
where w pq = =p| ˆ
w|q denote the one-particle integrals (1.46). To prove the equivalence of the two forms (for a specific electron number N ), let us inspect the matrix
elements of ˆ
W
with respect to the basis set of N -electron product states. Using the
anticommutator algebra established above, the result of applying ˆ
W
to a general
product state becomes
p,q
w pq c
†
p c q |q 1 . . . q N =
p
q i
w pq i c
†
p c q i |q 1 . . . q N
=
p
q i
w pq i |q 1 . . . q i−1 pq i+1 . . . q N
Note that here no overall sign change occurs because a pair of operators is moved
to the respective position in the product state. Since, according to the SC rules a1
and a2,