486
9 The Unification of Structure and Reaction Frameworks
=
J T
T | ⊗ ⊗N CM L CM J int J P M P |
J A
M A
ˆ
H T + ˆ
H int + ˆ
H CM
|Ψ
J
T
T
⊗ |N
CM L
CM J
int J
P M
P
J A
M A
= (E T + E int + E CM )
J T
T |Ψ
J
T
T
× ×N CM L CM J int J P M P |N
CM L
CM J
int J
P M
P = 0 ,
(9.191)
where the overlap between projectile states vanishes because a = a. As a
consequence, H
J A M A
cc
(N CM , N
CM ) = 0 when a = a and N CM , N
CM >
N CMmax .
D. Using previous results, one can write the matrix element H
J A M A
cc
(R CM , R
CM ) as
a sum of two terms:
H
J A M A
cc
(R CM , R
CM )
=
N CM ≤N CMmax
N
CM ≤N CMmax
H
J A M A
cc
(N CM , N
CM )U N CM (R CM )U N
CM
(R
CM )
+δ cc
N CM >N CMmax
(E T + E int + E CM ) U N CM (R CM )U N
CM
(R
CM ) . (9.192)
The sums in Eq. (9.192) involving N CM > N CMmax and N
CM > N CMmax can
be written as
N CM >N CMmax
(E T + E int + E CM ) U N CM (R CM )U N
CM
(R
CM )
=
N CM
(E T + E int + E CM ) U N CM (R CM )U N
CM
(R
CM )
−
N CM ≤N CMmax
(E T + E int + E CM ) U N CM (R CM )U N
CM
(R
CM ) , (9.193)
where the first sum in Eq. (9.193) can be expressed with Dirac delta’s due to
completeness properties of ˆ
H CM eigenstates:
N CM
(E T + E int + E CM ) U N CM (R CM )U N CM (R
CM )
=
¯
h 2
2M P
−
d 2
dR 2
CM
+
L CM (L CM + 1)
R 2
CM
+ E T + E int
×δ(R CM − R
CM ) + U
L CM
CM (R CM , R
CM ) ,
(9.194)
where U
L CM
CM (R CM , R
CM ) stands for the potential part of Eq. (9.90).