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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 Berry connection and curvature ： ウィキペディア英語版 Berry connection and curvature In physics, Berry connection and Berry curvature are related concepts, which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase. These concepts were introduced by Michael Berry in a paper published in 1984〔 〕 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics. Such phases have come to be known as Berry phases. ==Berry phase and cyclic adiabatic evolution== In quantum mechanics, the Berry phase arises in a cyclic adiabatic evolution. The quantum adiabatic theorem applies to a system whose Hamiltonian $H(\backslash mathbf\; R)$ depends on a (vector) parameter $\backslash mathbf\; R$ that varies with time $t$. If the $n$'th eigenvalue $\backslash varepsilon\_n(\backslash mathbf\; R)$ remains non-degenerate everywhere along the path and the variation with time ''t'' is sufficiently slow, then a system initially in the eigenstate $\backslash ,\; |n(\backslash mathbf\; R(0))\backslash rangle$ will remain in an instantaneous eigenstate $\backslash ,\; |n(\backslash mathbf\; R(t))\backslash rangle$ of the Hamiltonian $\backslash ,\; H(\backslash mathbf\; R(t))$, up to a phase, throughout the process. Regarding the phase, the state at time ''t'' can be written as〔 〕 :$$ |\Psi_n(t)\rangle =e^\, e^\, | n(\mathbf R(t))\rangle, where the second exponential term is the "dynamic phase factor." The first exponential term is the geometric term, with $\backslash gamma\_n$ being the Berry phase. By plugging into the time-dependent Schrödinger equation, it can be shown that :$$ \gamma_n(t)=i\int_0^t dt'\,\langle n(\mathbf R(t'))||n(\mathbf R(t'))\rangle=i\int_^ d\mathbf R\,\langle n(\mathbf R)|\nabla_|n(\mathbf R)\rangle, indicating that the Berry phase only depends on the path in the parameter space, not on the rate at which the path is traversed. In the case of a cyclic evolution around a closed path $\backslash mathcal\; C$ such that $\backslash mathbf\; R(T)=\backslash mathbf\; R(0)$, the closed-path Berry phase is :$$ \gamma_n=i\oint_ d\mathbf R\,\langle n(\mathbf R)|\nabla_|n(\mathbf R)\rangle. An example of physical system where an electron moves along a closed path is cyclotron motion (details are given in the page of Berry phase). Berry phase must be considered to obtain the correct quantization condition. 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「Berry connection and curvature」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.