翻訳と辞書 Words near each other ・ Semicassis inornata ・ Semicassis labiata ・ Semicassis pyrum ・ Semicassis royana ・ Semicassis saburon ・ Semicassis thachi ・ Semicassis thomsoni ・ Semice ・ Semicha in sacrifices ・ Semichem ・ Semichi Islands ・ Semicircle ・ Semicircle law ・ Semicircle law (Quantum Hall) ・ Semicircular canal ・ Semicircular potential well ・ Semiclassical ・ Semiclassical gravity ・ Semiclassical physics ・ Semiclivina ・ Semicollared flycatcher ・ Semicollared hawk ・ Semicollared puffbird ・ Semicolon ・ Semicomma ・ Semicomputable function ・ Semiconductor ・ Semiconductor (artists) ・ Semiconductor (disambiguation) ・ Semiconductor Bloch equations Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Semicircular potential well ： ウィキペディア英語版 Semicircular potential well In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The particle follows the path of a semicircle from $0$ to $\backslash pi$ where it cannot escape, because the potential from $\backslash pi$ to $2\; \backslash pi$ is infinite. Instead there is total reflection, meaning the particle bounces back and forth between $0$ to $\backslash pi$. The Schrödinger equation for a free particle which is restricted to a semicircle (technically, whose configuration space is the circle $S^1$) is :$-\backslash frac\backslash nabla^2\; \backslash psi\; =\; E\backslash psi\; \backslash quad\; (1)$ == Wave function == Using cylindrical coordinates on the 1-dimensional semicircle, the wave function depends only on the angular coordinate, and so :$\backslash nabla^2\; =\; \backslash frac\; \backslash frac\; \backslash quad\; (2)$ Substituting the Laplacian in cylindrical coordinates, the wave function is therefore expressed as :$-\backslash frac\; \backslash frac\; =\; E\backslash psi\; \backslash quad\; (3)$ The moment of inertia for a semicircle, best expressed in cylindrical coordinates, is $I\; \backslash \; \backslash stackrel\backslash \; \backslash iiint\_V\; r^2\; \backslash ,\backslash rho(r,\backslash phi,z)\backslash ,r\; dr\backslash ,d\backslash phi\backslash ,dz\; \backslash !$. Solving the integral, one finds that the moment of inertia of a semicircle is $I=m\; s^2$, exactly the same for a hoop of the same radius. The wave function can now be expressed as $-\backslash frac\; \backslash frac\; =\; E\backslash psi$, which is easily solvable. Since the particle cannot escape the region from $0$ to $\backslash pi$, the general solution to this differential equation is :$\backslash \; \backslash psi\; (\backslash phi)\; =\; A\; \backslash cos(m\; \backslash phi)\; +\; B\; \backslash sin\; (m\; \backslash phi)\; \backslash quad\; (4)$ Defining $m=\backslash sqrt\; \}$, we can calculate the energy as $E=\; \backslash frac$. We then apply the boundary conditions, where $\backslash psi$ and $\backslash frac$ are continuous and the wave function is normalizable: :$\backslash int\_^\; \backslash left|\; \backslash psi\; (\; \backslash phi\; )\; \backslash right|^2\; \backslash ,\; d\backslash phi\; =\; 1\backslash \; \backslash quad\; (5)$. Like the infinite square well, the first boundary condition demands that the wave function equals 0 at both $\backslash phi\; =\; 0$ and $\backslash phi\; =\; \backslash pi$. Basically :$\backslash \; \backslash psi\; (0)\; =\; \backslash psi\; (\backslash pi)\; =\; 0\; \backslash quad\; (6)$. Since the wave function $\backslash \; \backslash psi(0)\; =\; 0$, the coefficient A must equal 0 because $\backslash \; \backslash cos\; (0)\; =\; 1$. The wave function also equals 0 at $\backslash phi=\; \backslash pi$ so we must apply this boundary condition. Discarding the trivial solution where B=0, the wave function $\backslash \; \backslash psi\; (\backslash pi)\; =\; 0\; =\; B\; \backslash sin\; (m\; \backslash pi)$ only when m is an integer since $\backslash sin\; (n\; \backslash pi)\; =\; 0$. This boundary condition quantizes the energy where the energy equals $E=\; \backslash frac$ where m is any integer. The condition m=0 is ruled out because $\backslash psi\; =\; 0$ everywhere, meaning that the particle is not in the potential at all. Negative integers are also ruled out. We then normalize the wave function, yielding a result where $B=\; \backslash sqrt\}$. The normalized wave function is :$\backslash \; \backslash psi\; (\backslash phi)\; =\; \backslash sqrt\}\; \backslash sin\; (m\; \backslash phi)\; \backslash quad\; (7)$. The ground state energy of the system is $E=\; \backslash frac$. Like the particle in a box, there exists nodes in the excited states of the system where both $\backslash \; \backslash psi\; (\backslash phi)$ and $\backslash \; \backslash psi\; (\backslash phi)\; ^2$ are both 0, which means that the probability of finding the particle at these nodes are 0. 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「Semicircular potential well」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.