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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 Racah W-coefficient ： ウィキペディア英語版 Racah W-coefficient Racah's W-coefficients were introduced by Giulio Racah in 1942. These coefficients have a purely mathematical definition. In physics they are used in calculations involving the quantum mechanical description of angular momentum, for example in atomic theory. The coefficients appear when there are three sources of angular momentum in the problem. For example, consider an atom with one electron in an s orbital and one electron in a p orbital. Each electron has electron spin angular momentum and in addition the p orbital has orbital angular momentum (an s orbital has zero orbital angular momentum). The atom may be described by ''LS'' coupling or by ''jj'' coupling as explained in the article on angular momentum coupling. The transformation between the wave functions that correspond to these two couplings involves a Racah W-coefficient. Apart from a phase factor, Racah's W-coefficients are equal to Wigner's 6-j symbols, so any equation involving Racah's W-coefficients may be rewritten using 6-j symbols. This is often advantageous because the symmetry properties of 6-j symbols are easier to remember. Racah coefficients are related to recoupling coefficients by :$$ W(j_1j_2Jj_3;J_J_) \equiv \frac,j_3)J \rangle}+1)}}. Recoupling coefficients are elements of a unitary transformation and their definition is given in the next section. Racah coefficients have more convenient symmetry properties than the recoupling coefficients (but less convenient than the 6-j symbols). ==Recoupling coefficients== Coupling of two angular momenta $\backslash mathbf\_1$ and $\backslash mathbf\_2$ is the construction of simultaneous eigenfunctions of $\backslash mathbf^2$ and $J\_z$, where $\backslash mathbf=\backslash mathbf\_1+\backslash mathbf\_2$, as explained in the article on Clebsch–Gordan coefficients. The result is :$$ |(j_1j_2)JM\rangle = \sum_^ \sum_^ |j_1m_1\rangle |j_2m_2\rangle \langle j_1m_1j_2m_2|JM\rangle, where $J=|j\_1-j\_2|,\backslash ldots,j\_1+j\_2$ and $M=-J,\backslash ldots,J$. Coupling of three angular momenta $\backslash mathbf\_1$, $\backslash mathbf\_2$, and $\backslash mathbf\_3$, may be done by first coupling $\backslash mathbf\_1$ and $\backslash mathbf\_2$ to $\backslash mathbf\_$ and next coupling $\backslash mathbf\_$ and $\backslash mathbf\_3$ to total angular momentum $\backslash mathbf$: :$$ |((j_1j_2)J_j_3)JM\rangle = \sum_}^^ |(j_1j_2)J_M_\rangle |j_3m_3\rangle \langle J_M_j_3m_3|JM\rangle Alternatively, one may first couple $\backslash mathbf\_2$ and $\backslash mathbf\_3$ to $\backslash mathbf\_$ and next couple $\backslash mathbf\_1$ and $\backslash mathbf\_$ to $\backslash mathbf$: :$$ |(j_1,(j_2j_3)J_)JM \rangle = \sum_^ \sum_}^M_\rangle \langle j_1m_1J_M_|JM\rangle Both coupling schemes result in complete orthonormal bases for the $(2j\_1+1)(2j\_2+1)(2j\_3+1)$ dimensional space spanned by :$$ |j_1 m_1\rangle |j_2 m_2\rangle |j_3 m_3\rangle, \;\; m_1=-j_1,\ldots,j_1;\;\; m_2=-j_2,\ldots,j_2;\;\; m_3=-j_3,\ldots,j_3. Hence, the two total angular momentum bases are related by a unitary transformation. The matrix elements of this unitary transformation are given by a scalar product and are known as recoupling coefficients. The coefficients are independent of $M$ and so we have :$$ |((j_1j_2)J_j_3)JM\rangle = \sum_)JM \rangle \langle (j_1,(j_2j_3)J_)J |((j_1j_2)J_j_3)J\rangle. The independence of $M$ follows readily by writing this equation for $M=J$ and applying the lowering operator $J\_-$ to both sides of the equation. 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「Racah W-coefficient」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.