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In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations 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 and Paul Gordan, who encountered an equivalent problem in invariant theory. From a vector calculus perspective, the CG coefficients associated with the SO(3) group can be defined simply as integrals of spherical harmonics. The addition of spins in quantum-mechanical terms can be read directly from this approach. The formulas below use Dirac's bra–ket notation. From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found. To find numerical values for the coefficients a phase convention must be adopted. In this article, the Condon–Shortley phase convention is chosen. == Angular momentum operators == Angular momentum operators are self-adjoint operators , , and that satisfy the commutation relations : where is the Levi-Civita symbol. Together the three operators define a ''vector operator'' (also known as a spherical vector): : By developing this concept further, one can define another operator as the inner product of with itself: :. This is an example of a Casimir operator. One can also define ''raising'' () and ''lowering'' () operators (the so-called ladder operators): :. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Clebsch–Gordan coefficients」の詳細全文を読む スポンサード リンク
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