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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: : where is the mass of the particle, is the momentum operator, and the potential depends only on , 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 , and . 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 to determine the energies for the particular potential under discussion. == Structure of the eigenfunctions == The eigenstates of the system have the form : 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 : The differential equation which characterizes the function 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
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