B.3.d. Matrix Elements
Consider an Nparticle operator that is a sum of N 1particle operators, i.e.,
O
ˆ
(
N
)
(
x
1
⋯
x
N
)=
∑
j=1
N
O
ˆ
(
1
)
(
x
j
)
(B.52)
where
x
j
=(
q
j
,
p
j
,
σ
j
)
. Now, consider the matrix element
⟨
α
1
⋯
α
j
⋯
α
N

O
ˆ
(
1
)
(
x
j
)
α
′
1
⋯
α
′
j
⋯
α
′
N
⟩
=⟨
α
j

O
ˆ
(
1
)
(
x
)
α
′
j
⟩⟨
α
1
⋯⋯
α
N

α
′
1
⋯⋯
α
′
N
⟩
Thus, the matrix elements of
O
ˆ
(
N
)
are expected to be a sum of matrix elements of
O
ˆ
(
1
)
,
i.e.,
⟨
α
1
⋯
α
j
⋯
α
N

(
S/A
)
O
ˆ
(
N
)
(
x
1
⋯
x
N
)

α
′
1
⋯
α
′
j
⋯
α
′
N
⟩
(
S/A
)
∝
∑
jj'
⟨
α
j

O
ˆ
(
1
)
(
x
)
α
′
j
⟩
⟨
α
1
⋯
α
N

(
S/A
)
a
α
+
a
α'

α
′
1
⋯
α
′
N
⟩
(
S/A
)
Indeed, after some tedious counting of
states [see e.g., chapter 1 of Fetter & Walecka], it can be shown that if

α
1
⋯
α
j
⋯
α
N
⟩
(
S/A
)
=
n
0
⋯
n
α
⋯
n
∞
⟩

α
′
1
⋯
α
′
j
⋯
α
′
N
⟩
(
S/A
)
=
n
′
0
⋯
n
′
α
⋯
n
′
∞
⟩
then
⟨
α
1
⋯
α
j
⋯
α
N

(
S/A
)
O
ˆ
(
N
)
(
x
1
⋯
x
N
)

α
′
1
⋯
α
′
j
⋯
α
′
N
⟩
(
S/A
)
=⟨
n
0
⋯
n
α
⋯
n
∞

O
ˆ

n
′
0
⋯
n
′
α
⋯
n
′
∞
⟩
provided
O
ˆ
=
∑
α=0
∞
∑
α'=0
∞
⟨α
O
ˆ
(
1
)
(
x
)
α
′
⟩
a
α
+
a
α'
(B53,72)
Note that (B53) is valid for both bosons
and fermions, as well as for all N.
Similarly, for an Nparticle operator that is a sum of 2particle operators, i.e.,
O
ˆ
(
N
)
(
x
1
⋯
x
N
)=
1
2
∑
i=1
N
∑
j=1≠i
N
O
ˆ
(
2
)
(
x
i
,
x
j
)
=
∑
i=1
N
∑
j=1
i−1
O
ˆ
(
2
)
(
x
i
,
x
j
)
where
O
ˆ
(
2
)
(
x
i
,
x
j
)=
O
ˆ
(
2
)
(
x
j
,
x
i
)
,
the nrepresentation version is
O
ˆ
=
1
2
∑
α=0
∞
∑
β=0
∞
∑
α'=0
∞
∑
β'=0
∞
⟨
αβ

O
ˆ
(
2
)
(
x
1
,
x
2
)
α
′
β
′
⟩
a
α
+
a
β
+
a
β'
a
α'
(B.58,73)
Note that in
⟨
αβ

O
ˆ
(
2
)
(
x,
x
′
)
α
′
β
′
⟩
,
the states associated with the 1^{st} (2^{nd}) particle are a, a' (b, b ' ).
The particular order of the operators in
a
α
+
a
β
+
a
β'
a
α'
should be noted.