438
9 The Unification of Structure and Reaction Frameworks
of the H
J A M A
cc
(R CM , R
CM ) expansion for identical partitions and N CM , N
CM >
N CMmax . Show that this sum vanishes for different partitions.
D. Write H
J A M A
cc
(R CM , R
CM ) as a function of target states, center-of-mass,
and projectile intrinsic energies using the completeness properties of the
eigenstates of ˆ
H CM .
Due to the decoupling of the target and projectile at high energy, it is more
convenient to rewrite the Hamiltonian ˆ
H by introducing the target Hamiltonian ˆ
H T :
ˆ
H T = ˆ
T T + ˆ
U T; basis + ( ˆ
V res − ˆ
U 0 )
A−a ,
(9.116)
where ( ˆ
V res − ˆ
U 0 )
A−a is the part of ˆ
V res − ˆ
U 0 acting on the (A−a)-body target state,
and where ˆ
T T and ˆ
U T
basis are the target kinetic and potential parts of the Hamiltonian
ˆ
H , respectively. The action of target and projectile Hamiltonians ˆ
H T and ˆ
H P (see
Eq. (9.88)) on A-body states is performed by considering non-fully antisymmetrized
A-body states:
ˆ
H T (|Ψ T ⊗ |Ψ P ) = ( ˆ
H T |Ψ T ⊗ |Ψ P
(9.117)
ˆ
H P (|Ψ T ⊗ |Ψ P ) =
|Ψ T ⊗ ˆ
H P |Ψ P
.
(9.118)
Thus, one can write the Hamiltonian as
ˆ
H = ˆ
H T + ˆ
H P + ˆ
H TP ,
(9.119)
where ˆ
H TP = ˆ
H − ˆ
H T − ˆ
H P by definition.
Similarly to the one-nucleon projectile, the matrix elements H
J A M A
cc
(R CM , R
CM )
are expanded in a basis of center-of-mass harmonic oscillator states (see
Sect. 9.1.1.2):
H
J A M A
cc
(N CM
HO
i , N CM
HO
f ) =
N CMmax
N CM N CMf α i α f
c
(i)
α i
c
(f)
α f
SD α f | ˆ
H |SD α i
× ×N CM
HO
f |N CMf CM
HO
i |N CMi
−
N CM ≤N CMmax
(E T + E int + E CM )
× ×N CM
HO
f |N CM CM
HO
i |N CM , (9.120)
where E T is the energy of |Ψ
J T
T E int and E CM are the intrinsic and center-of-mass
energies of |N CM L CM J int J P M P , respectively, and the expansion coefficients c
(i)
α
and c
(f)
α are those of Eq. (9.108) for the input and output channels, respectively. The
9 The Unification of Structure and Reaction Frameworks
of the H
J A M A
cc
(R CM , R
CM ) expansion for identical partitions and N CM , N
CM >
N CMmax . Show that this sum vanishes for different partitions.
D. Write H
J A M A
cc
(R CM , R
CM ) as a function of target states, center-of-mass,
and projectile intrinsic energies using the completeness properties of the
eigenstates of ˆ
H CM .
Due to the decoupling of the target and projectile at high energy, it is more
convenient to rewrite the Hamiltonian ˆ
H by introducing the target Hamiltonian ˆ
H T :
ˆ
H T = ˆ
T T + ˆ
U T; basis + ( ˆ
V res − ˆ
U 0 )
A−a ,
(9.116)
where ( ˆ
V res − ˆ
U 0 )
A−a is the part of ˆ
V res − ˆ
U 0 acting on the (A−a)-body target state,
and where ˆ
T T and ˆ
U T
basis are the target kinetic and potential parts of the Hamiltonian
ˆ
H , respectively. The action of target and projectile Hamiltonians ˆ
H T and ˆ
H P (see
Eq. (9.88)) on A-body states is performed by considering non-fully antisymmetrized
A-body states:
ˆ
H T (|Ψ T ⊗ |Ψ P ) = ( ˆ
H T |Ψ T ⊗ |Ψ P
(9.117)
ˆ
H P (|Ψ T ⊗ |Ψ P ) =
|Ψ T ⊗ ˆ
H P |Ψ P
.
(9.118)
Thus, one can write the Hamiltonian as
ˆ
H = ˆ
H T + ˆ
H P + ˆ
H TP ,
(9.119)
where ˆ
H TP = ˆ
H − ˆ
H T − ˆ
H P by definition.
Similarly to the one-nucleon projectile, the matrix elements H
J A M A
cc
(R CM , R
CM )
are expanded in a basis of center-of-mass harmonic oscillator states (see
Sect. 9.1.1.2):
H
J A M A
cc
(N CM
HO
i , N CM
HO
f ) =
N CMmax
N CM N CMf α i α f
c
(i)
α i
c
(f)
α f
SD α f | ˆ
H |SD α i
× ×N CM
HO
f |N CMf CM
HO
i |N CMi
−
N CM ≤N CMmax
(E T + E int + E CM )
× ×N CM
HO
f |N CM CM
HO
i |N CM , (9.120)
where E T is the energy of |Ψ
J T
T E int and E CM are the intrinsic and center-of-mass
energies of |N CM L CM J int J P M P , respectively, and the expansion coefficients c
(i)
α
and c
(f)
α are those of Eq. (9.108) for the input and output channels, respectively. The