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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 Quantum pendulum ： ウィキペディア英語版 Quantum pendulum The quantum pendulum is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms as well as numerous other quantum phenomena. Though a pendulum not subject to the small-angle approximation has an inherent non-linearity, the Schrödinger equation for the quantized system can be solved relatively easily. ==Schrödinger Equation== Using Lagrangian theory from classical mechanics, one can develop a Hamiltonian for the system. A simple pendulum has one generalized coordinate (the angular displacement $\backslash phi$) and two constraints (the length of the string is constant and there is no motion along the z axis). The kinetic energy and potential energy of the system can be found to be as follows: :$T=\backslash frac\; m\; l^2\; \backslash dot^2$ :$U=m\; g\; l\; (1-\backslash cos(\backslash phi))$ This results in the Hamiltonian: :$\backslash hat\; =\; \backslash frac\; +\; m\; g\; l\; (1-\backslash cos(\backslash phi))$ The time-dependent Schrödinger equation for the system is as follows: :$i\; \backslash hbar\; \backslash frac\; =\; -\; \backslash frac\; \backslash frac\; +m\; g\; l\; (1-\backslash cos(\backslash phi))\; \backslash Psi$ One must solve the time-independent Schrödinger equation to find the energy levels and corresponding eigenstates. This is best accomplished by changing the independent variable as follows: :$\backslash eta\; =\; \backslash phi\; +\; \backslash pi$ :$E\; \backslash psi\; =\; -\; \backslash frac\; \backslash frac\; +m\; g\; l\; (1+\backslash cos(\backslash eta))\; \backslash psi$ This is simply Mathieu's equation where the solutions are Mathieu functions :$0\; =\; \backslash frac\; +(\backslash frac\; -\backslash frac\; -\backslash frac\; \backslash cos(\backslash eta))\; \backslash psi$ 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「Quantum pendulum」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.