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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 Holstein–Primakoff transformation ： ウィキペディア英語版 Holstein–Primakoff transformation One of the very important aspects of quantum mechanics is the occurrence of—in general—non-commuting operators which represent observables, quantities that we can measure. A standard example of a set of such operators are the three components of the angular momentum operators, which are crucial in many quantum systems. These operators are complicated, and we would like to be able to find a simpler representation, which can be used to generate approximate calculational schemes. The original〔T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 - 1113 (1940) http://link.aps.org/doi/10.1103/PhysRev.58.1098 〕 Holstein-Primakoff transformation in quantum mechanics is a mapping from the angular momentum operators to boson creation and annihilation operators. As can be seen from a paper with about 1000 citations, this method has found widespread applicability and has been extended in many different directions. There is a close link to other methods of boson mapping of operator algebras; in particular the Dyson-Maleev〔A. Klein and E. R. Marshalek, Boson realizations of Lie algebras with applications to nuclear physic,s http://link.aps.org/doi/10.1103/RevModPhys.63.375 〕 technique, and to a lesser extent the Schwinger mapping.〔J. Schwinger, Quantum theory of angular momentum, Academic Press, New York (1965)〕 There is a close link to the theory of (generalized) coherent states in Lie algebras. ==The basic technique== The basic idea can be illustrated for the classical example of the angular momentum operators of quantum mechanics. For any set of right-handed orthogonal axes we can define the components of this vector operator as $S\_x$, $S\_y$ and $S\_z$, which are mutually noncommuting, i.e., $\backslash left()=i\backslash hbar\; S\_z$ and cyclic permutations. In order to uniquely specify the states of a spin, we can diagonalise any set of commuting operators. Normally we use the SU(2) Casimir operators $S^2$ and $S\_z$, which leads to states with the quantum numbers $\backslash left|s,m\_s\backslash right\backslash rangle$: :$S^2\backslash left|s,m\_s\backslash right\backslash rangle=\backslash hbar^2\; s(s+1)\; \backslash left|s,m\_s\backslash right\backslash rangle,$ :$S\_z\backslash left|s,m\_s\backslash right\backslash rangle=\backslash hbar\; m\_s\backslash left|s,m\_s\backslash right\backslash rangle.$ The projection quantum number $m\_s$ takes on all the values $-s,-s+1,\; \backslash ldots,s-1,\; s$. We look at a single particle of spin $s$ (i.e., we look at a single irreducible representation of SU(2)). Now take the state with maximal projection $\backslash left|s,m\_s=\; +s\backslash right\backslash rangle$, the extremal weight state as a vacuum for a set of boson operators, and each subsequent state with lower projection quantum number as a boson excitation of the previous one, :$\backslash left|s,s-n\backslash right\backslash rangle\backslash mapsto\; \backslash frac\; \backslash sqrt\}\backslash ,\; a$ :$S\_-\; =\; \backslash hbar\; \backslash sqrt\; a^\backslash dagger\backslash ,\; \backslash sqrt\}$ :$S\_z\; =\; \backslash hbar(s\; -\; a^\backslash dagger\; a)$ The transformation is particularly useful in the case where $s$ is large, when the square roots can be expanded as Taylor series, to give an expansion in decreasing powers of $s$. 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「Holstein–Primakoff transformation」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.