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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 Particle in a spherically symmetric potential ： ウィキペディア英語版 Particle in a spherically symmetric potential An important kind of problem in quantum mechanics is that of a particle in a spherically symmetric potential, i.e., a potential that depends only on the distance between the particle and a defined center point. In particular, if the particle in question is an electron and the potential is derived from Coulomb's law, then the problem can be used to describe a hydrogen-like (one-electron) atom (or ion). In the general case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form: :$\backslash hat\; =\; \backslash frac\; +\; V(r)$ where $m\_0$ is the mass of the particle, $\backslash hat$ is the momentum operator, and the potential $V(r)$ depends only on $r$, the modulus of the radius vector r. The quantum mechanical wavefunctions and energies (eigenvalues) are found by solving the Schrödinger equation with this Hamiltonian. Due to the spherical symmetry of the system, it is natural to use spherical coordinates $r$, $\backslash theta$ and $\backslash phi$. When this is done, the time-independent Schrödinger equation for the system is separable, allowing the angular problems to be dealt with easily, and leaving an ordinary differential equation in $r$ to determine the energies for the particular potential $V(r)$ under discussion. == Structure of the eigenfunctions == The eigenstates of the system have the form :$\backslash psi(r,\; \backslash theta,\; \backslash phi)\; =\; R(r)\backslash Theta(\backslash theta)\backslash Phi(\backslash phi)\; \backslash ,\backslash ;$ in which the spherical polar angles θ and φ represent the colatitude and azimuthal angle, respectively. The last two factors of ψ are often grouped together as spherical harmonics, so that the eigenfunctions take the form :$\backslash psi(r,\; \backslash theta,\; \backslash phi)\; =\; R(r)Y\_(\backslash theta,\backslash phi).\backslash ,$ The differential equation which characterizes the function $R(r)\; \backslash ,\backslash ;$ is called the radial equation. 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「Particle in a spherically symmetric potential」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.