490
9 The Unification of Structure and Reaction Frameworks
The last equation, provided with ˆ
J
LM L
(q, r) + (qr/2) ˆ
J LM L (q, r), is derived
from the equality related to ˆ
J
LM L
(q, r) and the following equation:
qr ˆ
J LM L (q, r) = (qr)
2
J LM L (q, r)
= (q
2 r
2
int + q
2 R
2
CM + 2q
2 (r int · R CM )) J LM L (q, r) .
The first three equalities enter the magnetic operator M M
L,M L
(see Eq. (9.54)). The
first and third equalities, involving a gradient, are used in scalar products involving
the spin s and orbital angular momentum l. The second equality arises after the
evaluation of the action of gradients on Bessel functions and spherical harmonics
(see Eq. (9.55)). The fourth equation is the electric charge operator (see Eq. (9.53)).
Exercise VII.
A. Spin is independent of center-of-mass coordinates, so that spin-dependent suboperators act only on the intrinsic part of the wave function. Consequently, one
can apply the Wigner–Eckart theorem on the intrinsic and center-of-mass parts
of the wave function to calculate matrix elements. However, as orbital angular
momentum possesses both relative and center-of-mass parts, its associated
matrix elements cannot be calculated using this method. Equations provided
with spin-dependent sub-operators are then straightforward to obtain from the
previous results, contrary to that depending on l, which will be considered in the
following.
B. As l = r × p, one obtains the demanded equation by replacing r and p by their
expression in terms of r int , R CM , p int , and P CM (see Eqs. (9.135), (9.136)):
l = (r int + R CM ) ×
p int +
m
M P
P CM
= l int +
m
M P
L CM
+
m
M P
r int × P CM + R CM × p int .
The operators entering that expression of l depend on either relative or centerof-mass coordinates. Therefore, the matrix elements of these operators can be
separated in relative and center-of-mass parts and thus can be conveniently
calculated.
C. Expressions are straightforward to derive from the results of Exercise VI and the
decomposition of l in intrinsic and center-of-mass operators provided in B. For
9 The Unification of Structure and Reaction Frameworks
The last equation, provided with ˆ
J
LM L
(q, r) + (qr/2) ˆ
J LM L (q, r), is derived
from the equality related to ˆ
J
LM L
(q, r) and the following equation:
qr ˆ
J LM L (q, r) = (qr)
2
J LM L (q, r)
= (q
2 r
2
int + q
2 R
2
CM + 2q
2 (r int · R CM )) J LM L (q, r) .
The first three equalities enter the magnetic operator M M
L,M L
(see Eq. (9.54)). The
first and third equalities, involving a gradient, are used in scalar products involving
the spin s and orbital angular momentum l. The second equality arises after the
evaluation of the action of gradients on Bessel functions and spherical harmonics
(see Eq. (9.55)). The fourth equation is the electric charge operator (see Eq. (9.53)).
Exercise VII.
A. Spin is independent of center-of-mass coordinates, so that spin-dependent suboperators act only on the intrinsic part of the wave function. Consequently, one
can apply the Wigner–Eckart theorem on the intrinsic and center-of-mass parts
of the wave function to calculate matrix elements. However, as orbital angular
momentum possesses both relative and center-of-mass parts, its associated
matrix elements cannot be calculated using this method. Equations provided
with spin-dependent sub-operators are then straightforward to obtain from the
previous results, contrary to that depending on l, which will be considered in the
following.
B. As l = r × p, one obtains the demanded equation by replacing r and p by their
expression in terms of r int , R CM , p int , and P CM (see Eqs. (9.135), (9.136)):
l = (r int + R CM ) ×
p int +
m
M P
P CM
= l int +
m
M P
L CM
+
m
M P
r int × P CM + R CM × p int .
The operators entering that expression of l depend on either relative or centerof-mass coordinates. Therefore, the matrix elements of these operators can be
separated in relative and center-of-mass parts and thus can be conveniently
calculated.
C. Expressions are straightforward to derive from the results of Exercise VI and the
decomposition of l in intrinsic and center-of-mass operators provided in B. For