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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 Antisymmetrizer ： ウィキペディア英語版 Antisymmetrizer In quantum mechanics, an antisymmetrizer $\backslash mathcal$ (also known as antisymmetrizing operator〔P.A.M. Dirac, ''The Principles of Quantum Mechanics'', 4th edition, Clarendon, Oxford UK, (1958) p. 248〕 ) is a linear operator that makes a wave function of ''N'' identical fermions antisymmetric under the exchange of the coordinates of any pair of fermions. After application of $\backslash mathcal$ the wave function satisfies the Pauli principle. Since $\backslash mathcal$ is a projection operator, application of the antisymmetrizer to a wave function that is already totally antisymmetric has no effect, acting as the identity operator. ==Mathematical definition == Consider a wave function depending on the space and spin coordinates of ''N'' fermions: :$$ \Psi(1,2, \ldots, N)\quad\text \quad i \leftrightarrow (\mathbf_i, \sigma_i), where the position vector r''i'' of particle ''i'' is a vector in $\backslash mathbb^3$ and σi takes on 2''s''+1 values, where ''s'' is the half-integral intrinsic spin of the fermion. For electrons ''s'' = 1/2 and σ can have two values ("spin-up": 1/2 and "spin-down": −1/2). It is assumed that the positions of the coordinates in the notation for Ψ have a well-defined meaning. For instance, the 2-fermion function Ψ(1,2) will in general be not the same as Ψ(2,1). This implies that in general $\backslash Psi(1,2)-\; \backslash Psi(2,1)\; \backslash ne\; 0$ and therefore we can define meaningfully a ''transposition operator'' $\backslash hat\_$ that interchanges the coordinates of particle ''i'' and ''j''. In general this operator will not be equal to the identity operator (although in special cases it may be). A transposition has the parity (also known as signature) −1. The Pauli principle postulates that a wave function of identical fermions must be an eigenfunction of a transposition operator with its parity as eigenvalue :$$ \begin \hat_ \Psi\big(1,2,\ldots,i, \ldots,j,\ldots, N\big)& \equiv \Psi\big(\pi(1),\pi(2),\ldots,\pi(i), \ldots,\pi(j),\ldots, \pi(N)\big) \\ &\equiv \Psi(1,2,\ldots,j, \ldots,i,\ldots, N) \\ &= - \Psi(1,2,\ldots,i, \ldots,j,\ldots, N). \end Here we associated the transposition operator $\backslash hat\_$ with the permutation of coordinates ''π'' that acts on the set of ''N'' coordinates. In this case ''π'' = (''ij''), where (''ij'') is the cycle notation for the transposition of the coordinates of particle ''i'' and ''j''. Transpositions may be composed (applied in sequence). This defines a product between the transpositions that is associative. It can be shown that an arbitrary permutation of ''N'' objects can be written as a product of transpositions and that the number of transposition in this decomposition is of fixed parity. That is, either a permutation is always decomposed in an even number of transpositions (the permutation is called even and has the parity +1), or a permutation is always decomposed in an odd number of transpositions and then it is an odd permutation with parity −1. Denoting the parity of an arbitrary permutation ''π'' by (−1)''π'', it follows that an antisymmetric wave function satisfies :$$ \hat \Psi\big(1,2,\ldots, N\big) \equiv \Psi\big(\pi(1),\pi(2),\ldots, \pi(N)\big) = (-1)^\pi \Psi(1,2,\ldots, N), where we associated the linear operator $\backslash hat$ with the permutation π. The set of all ''N''! permutations with the associative product: "apply one permutation after the other", is a group, known as the permutation group or symmetric group, denoted by ''S''''N''. We define the antisymmetrizer as :$$ \mathcal \equiv \frac \sum_ (-1)^\pi \hat . 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「Antisymmetrizer」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.