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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 Kubo formula ： ウィキペディア英語版 Kubo formula The Kubo Formula is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation. Among the numerous applications of linear response formula, one can mention charge and spin susceptibilities of, for instance, electron systems due to external electric or magnetic fields. Responses to external mechanical forces or vibrations can also be calculated using the very same formula. ==The general Kubo formula== Consider a quantum system described by the (time independent) Hamiltonian $H\_0$. The expectation value of a physical quantity, described by the operator $\backslash hat$, can be evaluated as: ::$\backslash langle\; \backslash hat\backslash rangle=Tr()=\backslash sum\_n\; \backslash langle\; n\; |\; \backslash hat\; |n\; \backslash rangle\; e^$ ::$\backslash hat=e^=\backslash sum\_n\; |n\; \backslash rangle\backslash langle\; n\; |e^$ where $Z\_0=Tr()$ is the partition function. Suppose now that just above some time $t=t\_0$ an external perturbation is applied to the system. The perturbation is described by an additional time dependence in the Hamiltonian: $\backslash hat(t)=\backslash hat\_0+\backslash hat(t)\; \backslash theta\; (t-t\_0),$ where $\backslash theta\; (t)$ is the Heaviside function ( = 1 for positive times, =0 otherwise) and $\backslash hat\; V(t)$ is hermitian and defined for all ''t'', so that $\backslash hat\; H(t)$ has for positive $t-t\_0$ again a complete set of real eigenvalues $E\_n(t).$ But these eigenvalues may change with time. However, we can again find the time evolution of the density matrix $\backslash hat(t)$ rsp. of the partition function $Z(t)=Tr\; ((t)),$ to evaluate the expectation value of $\backslash langle\backslash hat\; A\backslash rangle\; =\; Tr\; (\backslash rho\; (t)\backslash ,\backslash hat\; A)/Tr\; ((t)).$  The time dependence of the states $|n(t)\; \backslash rangle$ is governed by the Schrödinger equation $i\backslash partial\_t|n(t)\; \backslash rangle=\backslash hat(t)|n(t)\; \backslash rangle\; ,$ which thus determines everything, corresponding of course to the Schrödinger picture. But since $\backslash hat(t)$ is to be regarded as a small perturbation, it is convenient to now use instead the interaction picture representation, $|\backslash hat\; n(t)\backslash rangle\; ,$ in lowest nontrivial order. The time dependence in this representation is given by $|n(t)\; \backslash rangle=e^|\backslash hat(t)\; \backslash rangle=e^\backslash hat(t,t\_0)|\backslash hat(t\_0)\; \backslash rangle\; ,$ where by definition for all t and $t\_0$ it is: $|\backslash hat(t\_0)\; \backslash rangle=e^|n(t\_0)\; \backslash rangle$ To linear order in $\backslash hat(t)$, we have $\backslash hat\; (t,t\_0)=1-i\backslash int\_^t\; dt\text{'}\backslash hat\; V(t\text{'})$. Thus one obtains the expectation value of $\backslash hat(t)$ up to linear order in the perturbation. ::$\backslash begin$ \langle \hat(t)\rangle &=& \langle \hat\rangle_0-i\int_^t dt'\sum_n e^ \langle n (t_0)| \hat(t)\hat(t')- \hat(t')\hat(t) |n(t_0) \rangle\\ &=& \langle \hat\rangle_0-i\int_^t dt'\langle ()\rangle_0 \end The brackets $\backslash langle\; \backslash rangle\; \_0$ mean an equilibrium average with respect to the Hamiltonian $H\_0\; .$ Therefore, although the result is of first order in the perturbation, it involves only the zeroth-order eigenfunctions, which is usually the case in perturbation theory and moves away all complications which otherwise might arise for $t\; >\; t\_0$. Here we have used an example, where the operators are bosonic operators, while for fermionic operators, the retarded functions are defined with anti-communtators instead of the usual (see Second quantization) 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「Kubo formula」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.