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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク 9-j symbol ： ウィキペディア英語版 9-j symbol In physics, Wigner's 9-j symbols were introduced by Eugene Paul Wigner in 1937. They are related to recoupling coefficients in quantum mechanics involving four angular momenta :$$ \sqrt \begin j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end = \langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 | ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle. ==Recoupling of four angular momentum vectors== Coupling of two angular momenta $\backslash mathbf\_1$ and $\backslash mathbf\_2$ is the construction of simultaneous eigenfunctions of $\backslash mathbf^2$ and $J\_z$, where $\backslash mathbf=\backslash mathbf\_1+\backslash mathbf\_2$, as explained in the article on Clebsch–Gordan coefficients. Coupling of three angular momenta can be done in several ways, as explained in the article on Racah W-coefficients. Using the notation and techniques of that article, total angular momentum states that arise from coupling the angular momentum vectors $\backslash mathbf\_1$, $\backslash mathbf\_2$, $\backslash mathbf\_4$, and $\backslash mathbf\_5$ may be written as :$$ | ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle. Alternatively, one may first couple $\backslash mathbf\_1$ and $\backslash mathbf\_4$ to $\backslash mathbf\_7$ and $\backslash mathbf\_2$ and $\backslash mathbf\_5$ to $\backslash mathbf\_8$, before coupling $\backslash mathbf\_7$ and $\backslash mathbf\_8$ to $\backslash mathbf\_9$: :$$ |((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle. Both sets of functions provide a complete, orthonormal basis for the space with dimension $(2j\_1+1)(2j\_2+1)(2j\_4+1)(2j\_5+1)$ spanned by :$$ |j_1 m_1\rangle |j_2 m_2\rangle |j_4 m_4\rangle |j_5 m_5\rangle, \;\; m_1=-j_1,\ldots,j_1;\;\; m_2=-j_2,\ldots,j_2;\;\; m_4=-j_4,\ldots,j_4;\;\;m_5=-j_5,\ldots,j_5. Hence, the transformation between the two sets is unitary and the matrix elements of the transformation are given by the scalar products of the functions. As in the case of the Racah W-coefficients the matrix elements are independent of the total angular momentum projection quantum number ($m\_9$): :$$ |((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle = \sum_\sum_ | ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle \langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 | ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle. 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「9-j symbol」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.