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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 Ehrenfest theorem ： ウィキペディア英語版 Ehrenfest theorem The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of the force on a massive particle moving in a scalar potential, Loosely speaking, one can thus say that "quantum mechanical expectation values obey Newton’s classical equations of motion". (This loose statement needs some caveats, see.〔(【引用サイトリンク】 first=Nicholas )〕) The Ehrenfest theorem is a special case of a more general relation between the expectation of any quantum mechanical operator and the expectation of the commutator of that operator with the Hamiltonian of the system where is some QM operator and is its expectation value. This more general theorem was not actually derived by Ehrenfest (it is due to Werner Heisenberg). It is most apparent in the Heisenberg picture of quantum mechanics, where it is just the expectation value of the Heisenberg equation of motion. It provides mathematical support to the correspondence principle. The reason is that Ehrenfest's theorem is closely related to Liouville's theorem of Hamiltonian mechanics, which involves the Poisson bracket instead of a commutator. Dirac's rule of thumb suggests that statements in quantum mechanics which contain a commutator correspond to statements in classical mechanics where the commutator is supplanted by a Poisson bracket multiplied by . This makes the operator expectation values obey corresponding classical equations of motion, provided the Hamiltonian is at most quadratic in the coordinates and momenta. Otherwise, the evolution equations still may hold approximately, provided fluctuations are small. == Derivation in the Schrödinger picture == Suppose some system is presently in a quantum state . If we want to know the instantaneous time derivative of the expectation value of , that is, by definition :$\backslash begin$ \frac\langle A\rangle &= \frac\int \Phi^ * A \Phi~dx^3 \\ &= \int \left( \frac \right) A\Phi~dx^3 + \int \Phi^ * \left( \frac\right) \Phi~dx^3 +\int \Phi^ * A \left( \frac \right) ~dx^3 \\ &= \int \left( \frac \right) A\Phi~dx^3 + \left\langle \frac\right\rangle + \int \Phi^ * A \left( \frac \right) ~dx^3 \end where we are integrating over all space. If we apply the Schrödinger equation, we find that :$\backslash frac\; =\; \backslash fracH\backslash Phi$ By taking the complex conjugate we find :$\backslash frac\; =\; -\backslash frac\backslash Phi^$ *H^ * = -\frac\Phi^ *H. 〔In bra–ket notation, $\backslash frac\backslash langle\; \backslash phi\; |x\backslash rangle\; =\backslash frac\backslash langle\; \backslash phi\; |\backslash hat|x\backslash rangle\; =\backslash frac\backslash langle\; \backslash phi\; |x\; \backslash rangle\; H=\backslash frac\backslash Phi^$ *H, where $\backslash hat$ is the Hamiltonian operator, and is the Hamiltonian represented in coordinate space (as is the case in the derivation above). In other words, we applied the adjoint operation to the entire Schrödinger equation, which flipped the order of operations for and .〕 Note , because the Hamiltonian is Hermitian. Placing this into the above equation we have :$\backslash frac\backslash langle\; A\backslash rangle\; =\; \backslash frac\backslash int\; \backslash Phi^$ * (AH-HA) \Phi~dx^3 + \left\langle \frac\right\rangle = \frac\langle ()\rangle + \left\langle \frac\right\rangle. Often (but not always) the operator is time independent, so that its derivative is zero and we can ignore the last term. 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「Ehrenfest theorem」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.