Consistent chiral three-nucleon interactions in ... - Theory Center

3 Mathematical basics
with arbitrary angular momenta quantum numbers j1 and j2 and corresponding
magnetic quantum numbers m1, m2. We couple the two angular momenta by in-
serting an identity operator in the coupled basis representation
1 =
j1,j2,J,M
|(j1, j2)JM〉〈(j1, j2)JM| (3.2)
with J as the coupled momentum and M its corresponding magnetic quantum
number. The parenthesis indicate that j1 couples with j2 to J. Using eq. (3.2) we
can relate the uncoupled states to the coupled ones via
|j1m1, j2m2〉 =
j ′ 1 ,j′ 2 ,J,M
|(j ′ 1 , j′ 2 )JM〉〈(j′ 1 , j′ 2 )JM|j1m1, j2m2〉 (3.3)
=
|(j1, j2)JM〉〈(j1, j2)JM|j1m1, j2m2〉 . (3.4)
J,M
The summations over j ′ 1 , j′ 2
vanish, as the state has to fulfill the eigenvalue equation
of the squared angular momentum operatorsj 2 1 ,j 2 2 before and after insertion of the
identity operator. The expansion coefficients on the right hand side of eq. (3.3) are
called Clebsch-Gordan coefficients, which will be denoted in the following by
j1 j2
m1 m2
yielding
J
≡ 〈(j1, j2)JM|j1m1, j2m2〉 , (3.5)
M
|j1m1, j2m2〉 =
J,M
j1 j2
m1 m2
J
|(j1, j2)JM〉 . (3.6)
M
Likewise, we can expand the coupled states in the uncoupled basis
|(j1, j2)JM〉 =
m1,m2
j1 j2
m1 m2
J
|j1m1, j2m2〉 . (3.7)
M
The Clebsch-Gordan coefficients can be chosen to be real numbers and they are
nonzero only if the triangular condition
|j1 − j2| ≤ J ≤ j1 + j2
as well as
(3.8)
m1 + m2 = M (3.9)
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