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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 Wigner–Eckart theorem ： ウィキペディア英語版 Wigner–Eckart theorem The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators on the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a Clebsch-Gordan coefficient. The name derives from physicists Eugene Wigner and Carl Eckart who developed the formalism as a link between the symmetry transformation groups of space (applied to the Schrödinger equations) and the laws of conservation of energy, momentum, and angular momentum.〔(Eckart Biography )– The National Academies Press〕 Mathematically, the Wigner–Eckart theorem is generally stated in the following way. Given a tensor operator $T^$ and two states of angular momenta $j$ and $j\text{'}$, there exists a constant $\backslash langle\; j\; \backslash |\; T^\; \backslash |\; j\text{'}\; \backslash rangle$ such that for all $m$, $m\text{'}$, and $q$, the following equation is satisfied: :$$ \langle j \, m | T^_q | j' \, m'\rangle = \langle j' \, m' \, k \, q | j \, m \rangle \langle j \| T^ \| j'\rangle Here, *$T^\_q$ is the -th component of the spherical tensor operator $T^$ of rank ,〔The parenthesized superscript provides a reminder of its rank. However, unlike , it need not be an actual index.〕 * $|j\; m\backslash rangle$ denotes an eigenstate of total angular momentum and its z-component , * $\backslash langle\; j\text{'}\; m\text{'}\; k\; q\; |\; j\; m\backslash rangle$ is the Clebsch–Gordan coefficient for coupling with to get , and * $\backslash langle\; j\; \backslash |\; T^\; \backslash |\; j\text{'}\; \backslash rangle$ denotes〔This is a special notation specific to the Wigner-Eckart theorem.〕 some value that does not depend on , , nor and is referred to as the reduced matrix element. In effect, the Wigner–Eckart theorem says that operating with a spherical tensor operator of rank on an angular momentum eigenstate is like adding a state with angular momentum ''k'' to the state. The matrix element one finds for the spherical tensor operator is proportional to a Clebsch–Gordan coefficient, which arises when considering adding two angular momenta. When stated another way, one can say that the Wigner–Eckart theorem is a theorem that tells you how vector operators behave in a subspace. Within a given subspace, a component of a vector operator will behave in a way proportional to the same component of the angular momentum operator. This definition is given in the book ''Quantum Mechanics'' by Cohen–Tannoudji, Diu and Laloe. ==Background and overview== 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「Wigner–Eckart theorem」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.