ClebschGordan coefficients are mathematical symbol used to integrate products of three spherical harmonics.
ClebschGordan coefficients commonly arise in applications involving the addition
of angular momentum in quantum mechanics. If products of more than three spherical harmonics are desired, then a generalization known
as Wigner 6jsymbols
or Wigner 9jsymbols
is used.
The ClebschGordan coefficients are variously written as , , ,
or . The ClebschGordan
coefficients are implemented in Mathematica as ClebschGordan[j1, m1, j2, m2, j, m].
The ClebschGordan coefficients are defined by
(1)

where , and satisfy
(2)

for .
Care is needed in interpreting analytic representations of
ClebschGordan coefficients since these coefficients are defined only on
measure zero sets. As a result, "generic"
symbolic formulas may not hold it certain cases, if at all. For
example, ClebschGordan[1, 0, j2, 0, 2, 0] evaluates to an
expression that is "generically" correct but not correct for the special
case , whereas ClebschGordan[1, 0, 1, 0, 2, 0] evaluates to the
correct value .
The coefficients are subject to the restrictions that be positive
integers or halfintegers, is an
integer, are positive or negative integers
or half integers,
(3)
 
(4)
 
(5)

and , ,
and (Abramowitz and Stegun
1972, p. 1006). In addition, by use of symmetry relations, coefficients may
always be put in the standard form
and .
The ClebschGordan coefficients are sometimes expressed using the related Racah Vcoefficients,
(6)

or Wigner 3jsymbols.
Connections among the three are
(7)

(8)

(9)

They have the symmetry
(10)

and obey the orthogonality relationships
(11)

(12)

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