Coupling of Angular Momenta Isospin Nucleon-Nucleon Interaction

Coupling of Angular Momenta
A fully commuting set of operators is thus only given by
Angular-momentum coupling: the new basis vectors |jm,j 1 ,j 2 > can be expanded
in the uncoupled basis states |j 1 m 1 j 2 m 2 > as
jmj
j1
j2
1
j2
= ∑ ∑ j1m1
j2m2
|
m = − j m = − j
1
1
2
2
Ĵ
2
,Ĵ
z
,Ĵ
2
1
and
The expansion coefficients are denoted as Clebsch-Gordon-coefficients:
jm
j
1
m
1
j
2
m
=
2
Ĵ
2
2
j jm
1
m1
j2m2
| jm = C j 1m1
j2m
2
The name derives from the German mathematicians Alfred Clebsch (1833–1872) and Paul Gordan (1837–1912)
(4)
Properties of Clebsch-Gordon-coefficients:
Selection rules: the coefficient is zero unless the quantum numbers fulfil the
two conditions
m = − j,...,
j
m 1
+ m 2
= m
j − j
j
1
− j2
≤ j ≤ j1
+ j
2
≤ j1
≤ j + j2
2
j − j ≤ j ≤ j + j
"triangular condition": the size of the total angular momentum is
restricted to those values that are allowed by the rules of vector addition,
where the vectors ,Ĵ and Ĵ form a triangle.
Ĵ1
2
1
2
1
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