经典轨道的封闭性和径向Schroinger方程的因式分解

Closeness of Classical Orbits and Factorization of Radial Schroinger Equation

摘要: 研究表明 ,保证经典轨道具有封闭性的 Bertrand定理可以进一步推广 ,在适当的角动量下 ,仍存在着非椭圆的闭合轨道 .对于屏蔽 Coulomb场,可获得广义Runge-Lenz矢量.这种轨道封闭性与径向 Schroodinger方程因式分解相对应. It is shown that for a particle with suitable angular momenta in the screened Coulomb potential or isotropic harmonic potential, there still exists closed orbits rather than ellipse, characterized by the conserved perihelion and aphelion vectors, i.e., extended Runge Lenz vector, which implies a higher dynamical symmetry than the geometrical symmetry SO 3. For the potential, factorization of the radial Schrdinger equation to produce raising and lowering operators is also pointed out.

Abstract: It is shown that for a particle with suitable angular momenta in the screened Coulomb potential or isotropic harmonic potential, there still exists closed orbits rather than ellipse, characterized by the conserved perihelion and aphelion vectors, i.e., extended Runge Lenz vector, which implies a higher dynamical symmetry than the geometrical symmetry SO 3. For the potential, factorization of the radial Schrdinger equation to produce raising and lowering operators is also pointed out.