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9 The Unification of Structure and Reaction Frameworks
Exercise VII
One will consider hereby the angular momentum dependent parts of the electric
and magnetic operators for the calculation of radiative capture cross sections of
clusters (see Exercise VI).
A. Let us firstly consider the parts of the electromagnetic Hamiltonian that depend
on spin s and r, but not on orbital angular momentum l. Explain why the use
of s in Eqs. (9.53) and (9.55) poses no problem, contrary to l. Using the latter
result, derive the spin-dependent operators of Eq. (9.55) in relative and centerof-mass coordinates:
∇
ˆ
J
LM L
(q, r)
· s
=
int ,L CM
A
L CM
int
∇ int
ˆ
J
int
(q, r int )
· s
⊗ J L CM (q, R CM )
L
M L
+
int ,L CM
A
L CM
int
∇ int
J int (q, r int )
· s
⊗ ˆ
J
L CM
(q, R CM )
L
M L
−
int ,L CM
A
L CM
int
∇ int
J int (q, r int )
· s
⊗ J L CM (q, R CM )
L
M L
q ˆ
J LM L (q, r) (s · e)
= q
int ,L CM
A
L CM
int
ˆ
J int (q, r int ) (s · e int ) ⊗ J L CM (q, R CM )
L
M L
+q
int ,L CM
A
L CM
int
J int (q, r int ) (s · E CM ) ⊗ ˆ
J L CM (q, R CM )
L
M L
,
where e = r/r.
B. One now concentrates on the l-dependent parts of the Hamiltonian of
Eq. (9.55).
Using Eqs. (9.135) and (9.136), show that the orbital angular momentum
writes
l = l int +
m
M P
L CM +
m
M P
(r int × P CM ) + R CM × p int .
Explain why this formulation is convenient to calculate l-dependent matrix
elements.
C. Express the operators
ˆ
J
LM L
(q, r)(l · s)