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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 Laplace expansion (potential) ： ウィキペディア英語版 Laplace expansion (potential) :''See also Laplace expansion of determinant''. In physics, the Laplace expansion of a 1/''r'' - type potential is applied to expand Newton's gravitational potential or Coulomb's electrostatic potential. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the interelectronic repulsion. The Laplace expansion is in fact the expansion of the inverse distance between two points. Let the points have position vectors r and r', then the Laplace expansion is :$$ \frac'|} = \sum_^\infty \frac \sum_^ (-1)^m \frac Y^_\ell(\theta, \varphi) Y^_\ell(\theta', \varphi'). Here r has the spherical polar coordinates (''r'', θ, φ) and r' has ( ''r, θ', φ'). Further ''r''< is min(''r'', ''r) and ''r''> is max(''r'', ''r). The function $Y^m\_$ is a normalized spherical harmonic function. The expansion takes a simpler form when written in terms of solid harmonics, :$$ \frac'|} = \sum_^\infty \sum_^ (-1)^m I^_\ell(\mathbf) R^_\ell(\mathbf')\quad\hbox\quad |\mathbf| > |\mathbf'|. ==Derivation== One writes :$$ \frac'|} = \frac\quad h \equiv \frac . We find here the generating function of the Legendre polynomials $P\_\backslash ell(\backslash cos\backslash gamma)$ : :$$ \frac^\infty h^\ell P_\ell(\cos\gamma). Use of the spherical harmonic addition theorem :$$ P_(\cos \gamma) = \frac \sum_^ (-1)^m Y^_(\theta, \varphi) Y^m_(\theta', \varphi') gives the desired result. 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「Laplace expansion (potential)」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.