翻訳と辞書 Words near each other ・ Foch, West Virginia ・ Foch-Allenby District ・ Foch-Gilttoyees Provincial Park ・ Fochabers ・ Fochriw ・ Fochteloo ・ Fochville ・ Focicchia ・ Focillidia ・ Focillistis ・ Focillodes ・ Focillopis ・ FOCJ ・ Fock ・ Fock matrix ・ Fock space ・ Fock state ・ Fockbek ・ Fockbek (Amt) ・ Focke ・ Focke Museum ・ Focke Rochen ・ Focke's wind tunnel (Bremen, Germany) ・ Focke-Achgelis ・ Focke-Achgelis Fa 223 ・ Focke-Achgelis Fa 225 ・ Focke-Achgelis Fa 269 ・ Focke-Achgelis Fa 284 ・ Focke-Achgelis Fa 330 ・ Focke-Wulf Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fock space ： ウィキペディア英語版 Fock space The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first introduced it in his 1932 paper ''Konfigurationsraum und zweite Quantelung''.〔V. Fock, ''Z. Phys''. 75 (1932), 622-647〕〔M.C. Reed, B. Simon, "Methods of Modern Mathematical Physics, Volume II", Academic Press 1975. Page 328.〕 Informally, a Fock space is the sum of a set of Hilbert spaces representing zero particle states, one particle states, two particle states, and so on. If the identical particles are bosons, the -particle states are vectors in a symmetrized tensor product of single-particle Hilbert spaces . If the identical particles are fermions, the -particle states are vectors in an antisymmetrized tensor product of single-particle Hilbert spaces . A general state in Fock space is a linear combination of n-particle states, one for each ''n''. Technically, the Fock space is (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space , :$F\_\backslash nu(H)=\backslash overlineS\_\backslash nu\; H^\}\; ~.$ Here $S\_\backslash nu$ is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic $(\backslash nu\; =\; +)$ or fermionic $(\backslash nu\; =\; -)$ statistics, and the overline represents the completion of the space. The bosonic (resp. fermionic) Fock space can alternatively be constructed as (the Hilbert space completion of) the symmetric tensors $F\_+(H)\; =\; \backslash overline$ (resp. alternating tensors $F\_-(H)\; =\; \backslash overline^$ * H}). For every basis for } there is a natural basis of the Fock space, the Fock states. == Definition == Fock space is the (Hilbert) direct sum of tensor products of copies of a single-particle Hilbert space $H$ :$F\_\backslash nu(H)=\backslash bigoplus\_^S\_\backslash nu\; H^\; =\backslash mathbb\; \backslash oplus\; H\; \backslash oplus\; \backslash left(S\_\backslash nu\; \backslash left(H\; \backslash otimes\; H\backslash right)\backslash right)\; \backslash oplus\; \backslash left(S\_\backslash nu\; \backslash left(\; H\; \backslash otimes\; H\; \backslash otimes\; H\backslash right)\backslash right)\; \backslash oplus\; \backslash ldots$ Here $\backslash mathbb$, the complex scalars, consists of the states corresponding to no particles, $H$ the states of one particle, $S\_\backslash nu\; (H\backslash otimes\; H)$ the states of two identical particles etc. A typical state in $F\_\backslash nu(H)$ is given by :$|\backslash Psi\backslash rangle\_\backslash nu=\; |\backslash Psi\_0\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Psi\_1\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Psi\_2\backslash rangle\_\backslash nu\; \backslash oplus\; \backslash ldots\; =\; a\_0\; |0\backslash rangle\; \backslash oplus\; |\backslash psi\_1\backslash rangle\; \backslash oplus\; \backslash sum\_\; a\_|\backslash psi\_,\; \backslash psi\_\; \backslash rangle\_\backslash nu\; \backslash oplus\; \backslash ldots$ where :$|0\backslash rangle$ is a vector of length 1, called the vacuum state and $\backslash ,a\_0\; \backslash in\; \backslash mathbb$ is a complex coefficient, :$|\backslash psi\_1\backslash rangle\; \backslash in\; H$ is a state in the single particle Hilbert space, :$|\backslash psi\_\; \backslash psi\_\; \backslash rangle\_\backslash nu\; =\; \backslash frac(|\backslash psi\_\backslash rangle\; \backslash otimes|\backslash psi\_\backslash rangle\; +\; (-1)^\backslash nu|\backslash psi\_\backslash rangle\backslash otimes|\backslash psi\_\backslash rangle)\; \backslash in\; S\_\backslash nu(H\; \backslash otimes\; H)$, and $a\_\; =\; \backslash nu\; a\_\; \backslash in\; \backslash mathbb$ is a complex coefficient :etc. The convergence of this infinite sum is important if $F\_\backslash nu(H)$ is to be a Hilbert space. Technically we require $F\_\backslash nu(H)$ to be the Hilbert space completion of the algebraic direct sum. It consists of all infinite tuples $|\backslash Psi\backslash rangle\_\backslash nu\; =\; (|\backslash Psi\_0\backslash rangle\_\backslash nu\; ,\; |\backslash Psi\_1\backslash rangle\_\backslash nu\; ,$ |\Psi_2\rangle_\nu, \ldots) such that the norm, defined by the inner product is finite : where the $n$ particle norm is defined by :$\backslash langle\; \backslash Psi\_n\; |\; \backslash Psi\_n\; \backslash rangle\_\backslash nu\; =\; \backslash sum\_a\_^$ *a_ \langle \psi_| \psi_ \rangle\cdots \langle \psi_| \psi_ \rangle i.e. the restriction of the norm on the tensor product $H^$ For two states :$|\backslash Psi\backslash rangle\_\backslash nu=\; |\backslash Psi\_0\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Psi\_1\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Psi\_2\backslash rangle\_\backslash nu\; \backslash oplus\; \backslash ldots\; =\; a\_0\; |0\backslash rangle\; \backslash oplus\; |\backslash psi\_1\backslash rangle\; \backslash oplus\; \backslash sum\_\; a\_|\backslash psi\_,\; \backslash psi\_\; \backslash rangle\_\backslash nu\; \backslash oplus\; \backslash ldots$, and :$|\backslash Phi\backslash rangle\_\backslash nu=|\backslash Phi\_0\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Phi\_1\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Phi\_2\backslash rangle\_\backslash nu\; \backslash oplus\; \backslash ldots\; =\; b\_0\; |0\backslash rangle\; \backslash oplus\; |\backslash phi\_1\backslash rangle\; \backslash oplus\; \backslash sum\_\; b\_|\backslash phi\_,\; \backslash phi\_\; \backslash rangle\_\backslash nu\; \backslash oplus\; \backslash ldots$ the inner product on $F\_\backslash nu(H)$ is then defined as :$\backslash langle\; \backslash Psi\; |\backslash Phi\backslash rangle\_\backslash nu:=\; \backslash sum\_n\; \backslash langle\; \backslash Psi\_n|\; \backslash Phi\_n\; \backslash rangle\_\backslash nu\; =\; a\_0^$ * b_0 + \langle\psi_1 | \phi_1 \rangle +\sum_a_^ *b_\langle \phi_|\psi_\rangle\langle\psi_| \phi_ \rangle_\nu + \ldots where we use the inner products on each of the $n$-particle Hilbert spaces. Note that, in particular the $n$ particle subspaces are orthogonal for different $n$. 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「Fock space」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.