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- In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where the type and the multiplicities of these irreducible representations are 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. Generalization to SU(3) of Clebsch–Gordan coefficients is useful because of their utility in characterizing hadronic decays, where a flavor-SU(3) symmetry exists (the eightfold way) that connects the three light quarks: up, down, and strange. (en)
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- 42134 (xsd:nonNegativeInteger)
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- A symmetric tensor operator analogous to the Laplace–Runge–Lenz vector for the Kepler problem may be defined,
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which commutes with the Hamiltonian,
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Since it commutes with the Hamiltonian , it represents 6−1=5 constants of motion.
It has the following properties,
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Apart from the tensorial trace of the operator, which is the Hamiltonian, the remaining 5 operators can be rearranged into their spherical component form as
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Further, the angular momentum operators are written in spherical component form as
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They obey the following commutation relations,
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The eight operators obey the same commutation relations as the infinitesimal generators of the group, detailed above.
As such, the symmetry group of Hamiltonian for a linear isotropic 3D Harmonic oscillator is isomorphic to group. (en)
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- Proof that the symmetry group of linear isotropic 3D Harmonic Oscillator is''' (en)
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- In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where the type and the multiplicities of these irreducible representations are 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. (en)
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- Clebsch–Gordan coefficients for SU(3) (en)
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