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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 D-matrix ： ウィキペディア英語版 Wigner D-matrix The Wigner D-matrix is a matrix in an irreducible representation of the groups SU(2) and SO(3). The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The matrix was introduced in 1927 by Eugene Wigner. == Definition of the Wigner D-matrix == Let ''Jx'', ''Jy'', ''Jz'' be generators of the Lie algebra of SU(2) and SO(3). In quantum mechanics these three operators are the components of a vector operator known as ''angular momentum''. Examples are the angular momentum of an electron in an atom, electronic spin, and the angular momentum of a rigid rotor. In all cases the three operators satisfy the following commutation relations, :$()\; =\; i\; J\_z,\backslash quad\; ()\; =\; i\; J\_y,\backslash quad\; ()\; =\; i\; J\_x,$ where ''i'' is the purely imaginary number and Planck's constant $\backslash hbar$ has been put equal to one. The operator :$J^2\; =\; J\_x^2\; +\; J\_y^2\; +\; J\_z^2$ is a Casimir operator of SU(2) (or SO(3) as the case may be). It may be diagonalized together with $J\_z$ (the choice of this operator is a convention), which commutes with $J^\; 2$. That is, it can be shown that there is a complete set of kets with :$J^2\; |jm\backslash rangle\; =\; j(j+1)\; |jm\backslash rangle,\backslash quad\; J\_z\; |jm\backslash rangle\; =\; m\; |jm\backslash rangle,$ where ''j'' = 0, 1/2, 1, 3/2, 2,... and ''m'' = -j, -j + 1,..., ''j''. For SO(3) the ''quantum number'' ''j'' is integer. A rotation operator can be written as :$\backslash mathcal(\backslash alpha,\backslash beta,\backslash gamma)\; =\; e^e^e^,$ where ''α'', ''β'', ''γ'' are Euler angles (characterized by the keywords: z-y-z convention, right-handed frame, right-hand screw rule, active interpretation). The Wigner D-matrix is a square matrix of dimension 2''j'' + 1 with general element :$D^j\_(\backslash alpha,\backslash beta,\backslash gamma)\; \backslash equiv$ \langle jm' | \mathcal(\alpha,\beta,\gamma)| jm \rangle = e^ d^j_(\beta)e^. The matrix with general element :$$ d^j_(\beta)= \langle jm' |e^ | jm \rangle is known as Wigner's (small) d-matrix. 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 D-matrix」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.