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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 Partial wave analysis ： ウィキペディア英語版 Partial wave analysis Partial wave analysis, in the context of quantum mechanics, refers to a technique for solving scattering problems by decomposing each wave into its constituent angular momentum components and solving using boundary conditions. == Preliminary scattering theory == The following description follows the canonical way of introducing elementary scattering theory. A steady beam of particles scatters off a spherically symmetric potential $V(r)$, which is short ranged so that for large distances $r\backslash to\backslash infty$, the particles behave like free particles. In principle, any particle should be described by a wave packet but we describe the scattering of a plane wave traveling along the z-axis $\backslash exp(ikz)$ instead, because wave packets are expanded in terms of plane waves and this is mathematically simpler. Because the beam is switched on for times long compared to the time of interaction of the particles with the scattering potential, a steady state is assumed. This means that the stationary Schrödinger equation for the wave function $\backslash Psi(\backslash vec\; x)$ representing the particle beam should be solved: :$\backslash left(+\; V(r)\backslash right)\backslash Psi(\backslash vec\; x)\; =\; E\backslash Psi(\backslash vec\; x)$ We make the following ''ansatz'': :$\backslash Psi(\backslash vec\; x)\; =\; \backslash Psi\_0(\backslash vec\; x)\; +\; \backslash Psi\_s(\backslash vec\; x)$ where $\backslash Psi\_0(\backslash vec\; x)\; \backslash propto\; \backslash exp(ikz)$ is the incoming plane wave and $\backslash Psi\_s(\backslash vec\; x)$ is a scattered part perturbing the original wave function. It is the asymptotic form of $\backslash Psi\_s(\backslash vec\; x)$ that is of interest, because observations near the scattering center (e.g. an atomic nucleus) are mostly not feasible and detection of particles takes place far away from the origin. At large distances, the particles should behave like free particles and $\backslash Psi\_s(\backslash vec\; x)$ should therefore be a solution to the free Schrödinger equation. This suggests that it should have a similar form to a plane wave, omitting any physically meaningless parts. We therefore investigate the plane wave expansion: :$e^\; =\; \backslash sum\_^\backslash infty\; (2\; \backslash ell\; +\; 1)\; i^\backslash ell\; j\_\backslash ell(k\; r)\; P\_\backslash ell(\backslash cos\; \backslash theta)$. The spherical Bessel function $j\_\backslash ell(kr)$ asymptotically behaves like :$j\_\backslash ell(kr)\; \backslash to\; \backslash frac\; 1\; r\; \backslash left(\backslash exp(i(kr-l\backslash pi/2))\; -\; \backslash exp(-i(kr-l\backslash pi/2))\backslash right).$ This corresponds to an outgoing and an incoming spherical wave. For the scattered wave function, only outgoing parts are expected. We therefore expect $\backslash Psi\_s(\backslash vec\; x)\backslash propto\; \backslash frac\; 1\; r\; \backslash exp(ikr)$ at large distances and set the asymptotic form of the scattered wave to :$\backslash Psi\_s(\backslash vec\; x)\backslash to\; f(\backslash theta,\; k)\backslash frac$ where $f(\backslash theta,\; k)$ is the so-called ''scattering amplitude'', which is in this case only dependent on the elevation angle $\backslash theta$ and the energy. In conclusion, this gives the following asymptotic expression for the entire wave function: :$\backslash Psi(\backslash vec\; x)\backslash to\; \backslash Psi^(\backslash vec\; x)\; =\; \backslash exp(ikz)\; +\; f(\backslash theta,\; k)\backslash frac$. 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「Partial wave analysis」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.