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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 cylindrical quadrupole ： ウィキペディア英語版 Quantum cylindrical quadrupole The solution to the Schrödinger equation, the wavefunction, describes the quantum mechanical properties of a particle on microscopic scales. Measurable quantities such as position, momentum and energy are all derived from the wavefunction.〔Davies, p. 1〕 $\backslash mathrm\backslash hbar\backslash frac\backslash psi(x,t)\; =\; -\backslash frac\backslash frac\backslash psi(x,t)\; +V(x)\backslash psi(x,t),$ where $\backslash hbar$ is the reduced Planck constant, $m$ is the mass of the particle, $\backslash mathrm$ is the imaginary unit and $t$ is time. One peculiar potential that can be solved exactly is when the electric quadrupole moment is the dominant term of an infinitely long cylinder of charge. It can be shown that the Schrödinger equation is solvable for a cylindrically symmetric electric quadrupole, thus indicating that the quadrupole term of an infinitely long cylinder can be quantized. In the physics of classical electrodynamics, it can be easily shown that the scalar potential and associated mechanical potential energy of a cylindrically symmetric quadrupole is as follows: :$\backslash mathbf\_\backslash mathrm\; =\; \backslash frac$ (SI units) :$\backslash mathbf\_\backslash mathrm\; =\; \backslash frac$ (SI units) Cylindrical symmetry should be used when solving the equation. The time independent Schrödinger equation becomes the following in cylindrical symmetry. $E\; \backslash psi(x)\; =\; -\backslash frac\backslash frac\; (s\; \backslash frac)\; \backslash psi(s,\backslash phi)-\backslash frac\backslash frac\backslash psi(s,\backslash phi)\; +\backslash frac\; \backslash psi(s,\backslash phi),$ Using the technique Separation of Variables, the above equation can be written as two ordinary differential equations in both the radial and azimuthal directions. The radial equation is Bessel's equation as can be seen below. If one changes variables to $x=\; k\; s$, Bessel's equation is exactly obtained. $\backslash frac\; \backslash frac\; (x\; \backslash frac)\; S(x)+(1-\backslash frac)\; S(x)=0$ ==Azimuthal equation== The azimuthal equation is Mathieu equation, is as follows: $\backslash frac\; \backslash Phi(\backslash phi)+(\backslash nu^2-\backslash frac\; Cos(\backslash phi))\; \backslash Phi()=0$ Since the canonical form of Mathieu's equation can be written as follows, it can be shown that $\backslash nu^2$ corresponds to a and $\backslash frac$ corresponds to q. :$\backslash frac+((2x))y=0.$ The solution of the Mathieu equation is expressed in terms of Mathieu cosine $C(a,q,x)$ Mathieu sine $S(a,q,x)$ for a unique a and q. This indicates that the quadrupole moment can be quantized in order of the Mathieu characteristic values $a\_n$ and $b\_n$. In general, Mathieu functions are not periodic. The term q must be that of a characteristic value in order for Mathieu functions to be periodic. Immediately, it can be shown that the solution of the radial equation highly depends on what characteristic values are seen in this case. 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 cylindrical quadrupole」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.