424
9 The Unification of Structure and Reaction Frameworks
standard with the Lawson method
|N CM = 0, L CM = 0, M CM = 0 |J int , M int .
(9.66)
In order to calculate center-of-mass harmonic oscillator many-body states efficiently, one introduces the ladder operator:
A
†
μ,CM =
Mω
2 ¯
h
R
(1)
μ − i
1
2M ¯
hω
P
(1)
μ ,
(9.67)
where μ = −1, 0, +1, and R
(1)
μ , P
(1)
μ represent the position and momentum of the
projectile in center-of-mass system, respectively,
R
(1)
μ =
1
a
a
i=1
r
(1)
i,μ
,
P
(1)
μ =
a
i=1
p
(1)
i,μ .
(9.68)
r
(1)
i,μ , and p
(1)
i,μ are, respectively, the position and impulsion of each nucleon of the
projectile in the center-of-mass system, and a is the number of nucleons in the
projectile. When one applies the A
†
μ,CM tensor operator on the |N CM L CM M CM
state, one increases the 2N CM + L CM by one harmonic oscillator center-of-mass
quantum:
2N CM + L CM → 2N CM + L CM + 1 ≡ 2N
CM + L
CM .
(9.69)
At this point, one couples the tensor operator A
†
CM acting on |N CM L CM J int , M int
to L
CM M
L
CM
A
†
CM |N CM L CM J int M int
L
CM
M
L CM
∝ |N
CM L
CM M
L
CM
J int M int .
(9.70)
Note that A
†
CM does not act on |J int M int . Therefore, one can build the set of states:
{|N CM L CM M L CM J int M int ,
(9.71)
because the harmonic oscillator degeneracy induced by A
†
μ,CM is lifted by the L CM
coupling. Then, using Clebsch–Gordan coefficients CM M L CM J int M int |J P M P
one couples the many-body state of Eq. (9.70) to J P . |N CM L CM M L CM J int
J P
M P
states
are calculated within harmonic oscillator shell model, so that they read
|N CM L CM M L CM J int
J P
M P
=
N
c
HO
N |SD N
HO .
(9.72)