Information AboutClebsch-gordan Coefficients |
| CATEGORIES ABOUT CLEBSCH-GORDAN COEFFICIENTS | |
| rotational symmetry | |
| representation theory of lie groups | |
| quantum mechanics | |
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In more mathematical terms, the CG coefficients are used in Representation Theory , particularly of Compact Lie Group s, to perform the explicit Direct Sum decomposition of the Tensor Product of two Irreducible Representation s into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly. The name derives from the German mathematicians Alfred Clebsch (1833-1872) and Paul Gordan (1837-1912), who encountered an equivalent problem in Invariant Theory . In terms of classical mathematics, the CG coefficients, or at least those associated to the group SO(3) , may be defined much more directly, by means of formulae for the multiplication of Spherical Harmonic s. The addition of spins in quantum-mechanical terms can be read directly from this approach. The formulas below use Dirac's Bra-ket Notation . FORMAL DEFINITION AND SOME RESULTS The Clebsch-Gordan coefficients are the numerical constants that express the Probability Amplitude for the Spin s with -projections to add to with projection | ||
| :<math>m | m_1+m_2</math> if <math>\langle j_1 j_2 m_1 m_2 j_1 j_2 j m
angle |
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| :<math>j 1j 2jm Angle | \sum_{m_1=-j_1}^{j_1}\sum_{m_2=-j_2}^{j_2}j_1j_2m_1m_2
angle\langle j_1j_2m_1m_2j_1j_2jm
angle</math> |
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| :<math>J \pmj 1j 2jm Angle | (J_{1\pm}+J_{2\pm})\sum_{m_1'=-j_1}^{j_1}\sum_{m_2'=-j_2}^{j_2}j_1j_2m_1'm_2'
angle\langle j_1j_2m_1'm_2'j_1j_2jm
angle</math> |
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| :::<math> | \sum_{m_1'}\sum_{m_2'}\left(\sqrt{(j_1\mp m_1')(j_1\pm m_1'+1)}j_1j_2m_1'\pm 1,m_2'
angle + \sqrt{(j_2\mp m_2')(j_2\pm m_2'+1)}j_1j_2m_1',m_2'\pm 1
angle
ight) |
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| \sqrt{(j\mp M)(j\pm M+1)}j 1j 2j,m\pm 1 Angle&& | &&\ |
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| ::<math> | \sum_{m_1'}\sum_{m_2'}\left(\sqrt{(j_1\mp m_1')(j_1\pm m_1'+1)}\langle j_1j_2m_1m_2j_1j_2m_1'\pm 1,m_2'
angle |
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| ::<math> | \sqrt{(j_1\mp m_1+1)(j_1\pm m_1)}\langle j_1j_2m_1\mp 1,m_2j_1j_2jm
angle |
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| ::<math> | \sqrt{(j_1\mp m_1+1)(j_1\pm m_1)}\langle j_1j_2m_1\mp 1,m_2j_1j_2j,m\mp 1
angle |
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