406
9 The Unification of Structure and Reaction Frameworks
where
rrj τ |u n j τ =
u n (r)
r
,
which implies for its associated particle creation operator
a
†
rrjτ =
n
u n (r)
r
a
†
nnj τ .
(9.5)
Hence, the expression of the A-body wave function becomes
ˆ
A |{|Ψ
J T
T ⊗ |rrj τ
J A
M A
=
n
u n (r)
r
ˆ
A |{|Ψ
J T
T ⊗ |u n j τ
J A
M A
=
n
u n (r)
r
{a
†
nnj τ |Ψ
J T
T
J A
M A
.
(9.6)
One will now expand the A-body wave function {a
†
nnj τ |Ψ
J T
T
J A
M A
, appearing in
Eq. (9.6), in a complete basis of A-body Slater determinants
{a
†
nnj τ |Ψ
J T
T
J A
M A
=
α
c α |SD α .
(9.7)
For that, one uses the expansion of |Ψ
J T ,M T
T
in (A − 1)-body Slater determinants,
obtained from the diagonalization of the Gamow shell model Hamiltonian H
|Ψ
J T ,M T
T
=
β
b
A−1
β
|SD β
A−1 ,
(9.8)
where the (A − 1)-body character of expansion coefficients and Slater determinants
is explicitly written. Applying the creation operator defined in Eq. (9.5) on Eq. (9.8),
one obtains the uncoupled fully antisymmetrized A-body wave function:
{a
†
nnj mτ |Ψ
J T ,M T
T
} =
β
b
A−1
β
a
†
nnj mτ |SD β
A−1
=
α
(−1)
ϕ α b α |SD α ,
(9.9)
where m is the angular momentum projection of the projectile state, verifying
M T + m = M A , and the Slater determinants |SD α = (−1) ϕ α a
†
nnj mτ |SD β A−1
are A-body basis functions, in which a (−1) ϕ α rearrangement phase appears. Note
that all A-body states, a
†
nnj mτ |SD β A−1 , for which |u n j mτ ∈ |SD β A−1 , vanish
identically due to the Pauli principle. The A-body wave function expansion of
Eq. (9.7) can be determined by coupling the wave function defined in Eq. (9.9) to
a given angular momentum using Clebsch–Gordan coefficients.