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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 Zitterbewegung ： ウィキペディア英語版 Zitterbewegung Zitterbewegung ("trembling motion" from German) is a hypothetical rapid motion of elementary particles, in particular electrons, that obey the Dirac equation. The existence of such motion was first proposed by Erwin Schrödinger in 1930 as a result of his analysis of the wave packet solutions of the Dirac equation for relativistic electrons in free space, in which an interference between positive and negative energy states produces what appears to be a fluctuation (at the speed of light) of the position of an electron around the median, with an angular frequency of $2\; m\; c^2\; /\; \backslash hbar$, or approximately 1.6 radians per second. A re-examination of Dirac theory, however, shows that interference between positive and negative energy states may not be a necessary criterion for observing zitterbewegung. Zitterbewegung of a free relativistic particle has never been observed. However it has been ''simulated'' twice. First, with a trapped ion, by putting it in an environment such that the non-relativistic Schrödinger equation for the ion has the same mathematical form as the Dirac equation (although the physical situation is different). Then, in 2013, it was simulated in a setup with Bose-Einstein condensates. ==Theory== The time-dependent Dirac equation :$H\; \backslash psi\; (\backslash mathbf,t)\; =\; i\; \backslash hbar\; \backslash frac\; (\backslash mathbf,t)$ where $H$ is the Dirac Hamiltonian for an electron in free space :$H\; =\; \backslash left(\backslash alpha\_0\; mc^2\; +\; \backslash sum\_^3\; \backslash alpha\_j\; p\_j\; \backslash ,\; c\backslash right)$ in the Heisenberg picture implies that any operator Q obeys the equation :$-i\; \backslash hbar\; \backslash frac\; (t)=\; \backslash left(H\; ,\; Q\; \backslash right)\; \backslash ,.$ In particular, the time-dependence of the position operator is given by :$\backslash hbar\; \backslash frac\; (t)=\; i\backslash left(H\; ,\; x\_k\; \backslash right)\; =\; \backslash hbar\; c\backslash alpha\_k$ where $\backslash alpha\_k\; \backslash equiv\; \backslash gamma\_0\; \backslash gamma\_k$. The above equation shows that the operator $\backslash alpha\_k$ can be interpreted as the kth component of a "velocity operator". To add time-dependence to $\backslash alpha\_k$, one implements the Heisenberg picture, which says :$\backslash alpha\_k\; (t)\; =\; e^\backslash alpha\_k\; e^$ The time-dependence of the velocity operator is given by :$\backslash hbar\; \backslash frac\; (t)=\; i\backslash left(H\; ,\; \backslash alpha\_k\; \backslash right)\; =\; 2(i\; \backslash gamma\_k\; m\; -\; \backslash sigma\_p^l)\; =\; 2i(p\_k-\backslash alpha\_kH)$ where $\backslash sigma\_\; \backslash equiv\; \backslash frac()$. Now, because both $p\_k$ and $H$ are time-independent, the above equation can easily be integrated twice to find the explicit time-dependence of the position operator. First: :$\backslash alpha\_k\; (t)\; =\; (\backslash alpha\_k\; (0)\; -\; c\; p\_k\; H^)\; e^\; +\; c\; p\_k\; H^$ Then: :$x\_k(t)\; =\; x\_k(0)\; +\; c^2\; p\_k\; H^\; t\; +\; i\; \backslash hbar\; c\; H^\; (\; \backslash alpha\_k\; (0)\; -\; c\; p\_k\; H^\; )\; (\; e^\; -\; 1\; )$ where $x\_k(t)$ is the position operator at time $t$. The resulting expression consists of an initial position, a motion proportional to time, and an unexpected oscillation term with an amplitude equal to the Compton wavelength. That oscillation term is the so-called "Zitterbewegung". Interestingly, the "Zitterbewegung" term vanishes on taking expectation values for wave-packets that are made up entirely of positive- (or entirely of negative-) energy waves. This can be achieved by taking a Foldy Wouthuysen transformation. Thus, we arrive at the interpretation of the "Zitterbewegung" as being caused by interference between positive- and negative-energy wave components. 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「Zitterbewegung」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.