Saturday, 23 July 2011

Clebsch–Gordan coefficients

Clebsch-Gordan coefficients are mathematical symbol used to integrate products of three spherical harmonics. Clebsch-Gordan 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 6j-symbols or Wigner 9j-symbols is used.
The Clebsch-Gordan coefficients are variously written as C_(m_1m_2)^j, C_(m_1m_2m)^(j_1j_2j), (j_1j_2m_1m_2|j_1j_2jm), or <j_1j_2m_1m_2|j_1j_2jm>. The Clebsch-Gordan coefficients are implemented in Mathematica as ClebschGordan[{j1, m1}, {j2, m2}, {j, m}].
The Clebsch-Gordan coefficients are defined by
 Psi_(JM)=sum_(M=M_1+M_2)C_(M_1M_2)^JPsi_(M_1M_2),
(1)
where J=J_1+J_2, and satisfy
 (j_1j_2m_1m_2|j_1j_2jm)=0
(2)
for m_1+m_2!=m.
Care is needed in interpreting analytic representations of Clebsch-Gordan 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 j_2=1, whereas ClebschGordan[{1, 0}, {1, 0}, {2, 0}] evaluates to the correct value sqrt(2/3).
The coefficients are subject to the restrictions that (j_1,j_2,j) be positive integers or half-integers, j_1+j_2+j is an integer, (m_1,m_2,m) are positive or negative integers or half integers,
j_1+j_2-j>=0
(3)
j_1-j_2+j>=0
(4)
-j_1+j_2+j>=0,
(5)
and -|j_1|<=m_1<=|j_1|, -|j_2|<=m_2<=|j_2|, and -|j|<=m<=|j| (Abramowitz and Stegun 1972, p. 1006). In addition, by use of symmetry relations, coefficients may always be put in the standard form j_1<j_2<j and m>=0.
The Clebsch-Gordan coefficients are sometimes expressed using the related Racah V-coefficients,
 V(j_1j_2j;m_1m_2m)
(6)
or Wigner 3j-symbols. Connections among the three are
 (j_1j_2m_1m_2|j_1j_2jm)=(-1)^(m+j_1-j_2)sqrt(2j+1)(j_1 j_2 j; m_1 m_2 -m)
(7)
 (j_1j_2m_1m_2|j_1j_2jm)=(-1)^(j+m)sqrt(2j+1)V(j_1j_2j;m_1m_2-m)
(8)
 V(j_1j_2j;m_1m_2m)=(-1)^(-j_1+j_2+j)(j_1 j_2 j_1; m_2 m_1 m_2).
(9)
They have the symmetry
 (j_1j_2m_1m_2|j_1j_2jm)=(-1)^(j_1+j_2-j)(j_2j_1m_2m_1|j_2j_1jm),
(10)
and obey the orthogonality relationships
 sum_(j,m)(j_1j_2m_1m_2|j_1j_2jm)(j_1j_2jm|j_1j_2m_1^'m_2^')=delta_(m_1m_1^')delta_(m_2m_2^')
(11)
 sum_(m_1,m_2)(j_1j_2m_1m_2|j_1j_2jm)(j_1j_2j^'m^'|j_1j_2m_1m_2)=delta_(jj^')delta_(mm^').
(12)

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