Clebsch-Gordan Theory / Coefficients| Clebsch-Gordan Theory / Coefficients |
In General
> s.a. angular momentum; group representations;
SU(2) [6j, 10j symbols].
* Idea: Mathematical
symbols used to integrate products of spherical harmonics, or add
angular momenta in quantum mechanics.
$ Def: The coefficients
C jm1,
m2
= \(\langle\) j1
j2
m1
m2
| j1
j2,
j m\(\rangle\), with
|j, m\(\rangle\) = ∑m1+m2=m \(\langle\) j1 j2 m1 m2 | j1 j2, j m\(\rangle\) |m1 m2\(\rangle\) , j1 + j2 = j.
* Properties:
\(\langle\) j1 m1, j2 m2 | j m\(\rangle\) = (−1) j1+j2−j \(\langle\) j2 m2, j1 m1 | j m \(\rangle\) .
* Relationships: In terms of 6j-symbols,
\[ \langle\, j_{_1}\,j_{_2}m_{_1}m_{_2}\vert\, j_{_1}j_{_2},j\,m\, \rangle
= (-1)^{-j_1+j_2-m}\,\sqrt{\vphantom{\sqrt1}2j+1}
\left(\matrix{j_{_1} & j_{_2} & j \cr m_{_1} & m_{_2} & -m}\right) .\]
* Theorem: πj ⊗ πj' = πj+j' ⊕ πj+j'−1 ⊕ ··· ⊕ π| j−j' |.
References
@ Calculation:
Klink & Wickramasekara EJP(10) [simple method];
Horst & Reuter CPC(11)-a1011 [CleGo computer package];
Ibort et al a1610 [numerical algorithm].
@ Asymptotics: de Guise & Rowe JMP(98),
Reinsch & Morehead JMP(99).
@ For SU(3): Coleman JMP(64);
Grigorescu SCF(84)mp/00;
Prakash & Sharatchandra JMP(96),
Rowe & Repka JMP(97) [and SU(2)];
Rowe & Bahri JMP(00).
@ Other groups: Asherova et al PAN(01)m.QA [Uq(su(3))];
Wu mp/06 [permutation group].
> Online resources:
see MathWorld page;
Wikipedia page
and table.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 4 oct 2016