B.2.2. AntiSymmetrized Momentum Eigenstates for
FermiDirac Particles
An antisymmetrized state can be
constructed as

k
1
,⋯,
k
N
⟩
(
−
)
=
∑
P
(
−
)
P
P
k
1
,⋯,
k
N
⟩
= Sum of all N! signed permutations
of the
k
i
's in

k
1
,⋯,
k
N
⟩
(B.25)
For example,
∑
P
(
−
)
P
P
k
1
,
k
2
,
k
3
⟩
=
k
1
,
k
2
,
k
3
⟩+
k
2
,
k
3
,
k
1
⟩+
k
3
,
k
1
,
k
2
⟩
−
k
2
,
k
1
,
k
3
⟩−
k
3
,
k
2
,
k
1
⟩−
k
1
,
k
3
,
k
2
⟩
(B.26)
Now, if a momentum
k
α
appears twice, permutation between only them
will give rise to 2 identical terms with opposite sign. Hence, the sum vanishes identically.
For example, setting
k
1
=
k
2
in (B.26) gives
∑
P
(
−
)
P
P
k
1
,
k
1
,
k
3
⟩
=
k
1
,
k
1
,
k
3
⟩+
k
1
,
k
3
,
k
1
⟩+
k
3
,
k
1
,
k
1
⟩
−
k
1
,
k
1
,
k
3
⟩−
k
3
,
k
1
,
k
1
⟩−
k
1
,
k
3
,
k
1
⟩
=0
The orthonormality of the 1particle states
implies
⟨
k
a
,
k
b
,⋯,
k
l

k
a'
,
k
b'
,⋯,
k
l'
⟩=
δ
aa'
δ
bb'
⋯
δ
ll'
Thus, for a given permutation
P(
k
1
,⋯,
k
N
)
,
we have
⟨
P(
k
1
,⋯,
k
N
)

k
1
,⋯,
k
N
⟩
(
−
)
=1
so that
⟨
k
1
,⋯,
k
N

k
1
,⋯,
k
N
⟩
(
−
)
=N!
An orthonormal set of antisymmetrized
states is therefore

k
1
,⋯,
k
N
⟩
(
A
)
=
1
N!

k
1
,⋯,
k
N
⟩
(
−
)
=
1
N!
∑
P
(
−
)
P
P
k
1
,⋯,
k
N
⟩
(B.25)
with
⟨
k
1
,⋯,
k
N

k
1
,⋯,
k
N
⟩
(
A
)
=1
(B.27)
Now, for a fixed
k
1
,⋯,
k
N
,
all the
N!
distinct permutations
P(
k
1
,⋯,
k
N
)
give rise to only 1 distinct antisymmetrized
state, i.e.,

k
1
,⋯,
k
N
⟩
(
A
)
=
(
−
)
P
P

k
1
,⋯,
k
N
⟩
(
A
)
. Therefore, in a sum
∑
k
1
,⋯,
k
N

k
1
⋯
k
N
⟩
(
A
)
,
each distinct antisymmetrized state will appear
N!
times.
Since the completeness relation involves a sum with each distinct
orthonormal state counted once, we have
1
N!
∑
k
1
⋯
k
N

k
1
⋯
k
N
⟩
(
A
)
⟨
k
1
⋯
k
N

=
1
ˆ
(
A
)
(B.28)
The wave function
⟨
r
1
,⋯,
r
N

k
1
,⋯,
k
N
⟩
(
A
)
can be written as
⟨
r
1
,⋯,
r
N

k
1
,⋯,
k
N
⟩
(
A
)
=
1
N
det
⟨
r
1

k
1
⟩
⋯
⟨
r
1

k
N
⟩
⋮
⟨
r
i

k
j
⟩
⋮
⟨
r
N

k
1
⟩
⋯
⟨
r
N

k
N
⟩

(B.29)
which is called the Slater determinant.