论文标题
平面$ 1/ρ^{2} $在多个坐标系统中的可分离性
Separability of the Planar $1/ρ^{2}$ Potential In Multiple Coordinate Systems
论文作者
论文摘要
对于许多特殊的哈密顿人，可以通过在多个坐标系统中的变量分离来找到Schrödinger方程的解决方案。涉及的一类电位包括许多重要例子，包括各向同性谐波振荡器和库仑电位。乘以可分离的汉密尔顿人具有许多有趣的特征，包括在其绑定状态光谱中的“偶然”变性，以及通常始终关闭的经典界面轨道。我们研究了另一个潜力，在圆柱和抛物线坐标中，Schrödinger方程都可以分离：A $ z $ - 独立$ V \ propto 1/ρ^{2} = 1/（x^{2}+y^{2}+y^{2}）$在三个方面。该电位关闭中的所有持久，绑定的经典轨道，因为所有其他具有负能量的轨道均以$ρ= 0 $落在中心。当在抛物线坐标中分离时，schrödinger方程将三个单独的方程式分成三个方程，其中两个等于库仑电势中的径向方程 - 一个具有有吸引力的电位的方程式，另一个具有同样强的排斥力。
With a number of special Hamiltonians, solutions of the Schrödinger equation may be found by separation of variables in more than one coordinate system. The class of potentials involved includes a number of important examples, including the isotropic harmonic oscillator and the Coulomb potential. Multiply separable Hamiltonians exhibit a number of interesting features, including "accidental" degeneracies in their bound state spectra and often classical bound state orbits that always close. We examine another potential, for which the Schrödinger equation is separable in both cylindrical and parabolic coordinates: a $z$-independent $V\propto 1/ρ^{2}=1/(x^{2}+y^{2})$ in three dimensions. All the persistent, bound classical orbits in this potential close, because all other orbits with negative energies fall to the center at $ρ=0$. When separated in parabolic coordinates, the Schrödinger equation splits into three individual equations, two of which are equivalent to the radial equation in a Coulomb potential---one equation with an attractive potential, the other with an equally strong repulsive potential.
