9.2 Generalization to Many-Nucleon Projectiles
437
channels, denoted by c,
|Ψ
J A
M A
=
c
+∞
0
u c (R CM )
R CM
|(c, R CM )
J A
M A
R CM
2 dR CM ,
(9.112)
with
|(c, R CM ) = ˆ
A |{|Ψ
J T
T ⊗ |R CM L CM J int J P M P
J A
M A
,
where the spin target J and the spin projectile J P are coupled to J A . The channel
index c stands for the {A − a, J T ; a, L CM , J int , J P } quantum numbers. u c (R CM )
is the radial amplitude of the c channel to be determined. The coupled-channel
equations can then be formally derived from the Schrödinger equation: H |Ψ
J A
M A
=
E |Ψ
J A
M A
, as
c
+∞
0
H cc (R CM , R
CM ) − EN cc (R CM , R
CM )
u c (R CM ) = 0 ,
(9.113)
with
H cc (R CM , R
CM ) = R CM R
CM (c, R CM )| ˆ
H |(c
, R
CM )
(9.114)
N cc (R CM , R
CM ) = R CM R
CM (c, R CM )|(c
, R
CM ) .
(9.115)
Exercise IV
One will derive the coupled-channel Hamiltonian matrix elements involving
cluster projectiles (see Eq. (9.120)).
A. Expand H
J A M A
cc
(R CM , R
CM ) using the eigenbasis of ˆ
H CM . One will use the
notation |N CM L CM J int J P M P and |N
CM L CM J int J P M P for the eigenbasis
expanding the R CM -dependent and R
CM -dependent channels, respectively.
Write the obtained expansion as four different sums, involving N CM ≤
N CMmax , N CM > N CMmax , N
CM ≤ N CMmax and N
CM > N CMmax .
B. Notice that the sum of the expansion function of N CM , N
CM ≤ N CMmax can
be conveniently calculated. Show that ˆ
H = ˆ
H T + ˆ
H CM + ˆ
H int + ˆ
A ˆ
H TP ˆ
A if
N CM > N CMmax or N
CM > N CMmax . Deduce that the non-vanishing sums of
the expansion are those for which N CM , N
CM ≤ N CMmax and N CM , N
CM >
N CMmax .
C. One will now consider input and output channels whose projectiles can bear
different numbers of nucleons, denoted as a for the input channel and a for
the output channel. They will be deemed as identical/different partitions if the
number of nucleons of these channels is identical/different. Calculate the sum