With a number of special Hamiltonians, solutions of the Schrodinger 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 Schrodinger equation is separable in both cylindrical and parabolic coordinates: a $z$-independent $V\propto 1/\rho^{2}=1/(x^{2}+y^{2})$ 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 $\rho=0$. When separated in parabolic coordinates, the Schrodinger 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.

中文翻译：

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多坐标系中平面 1/ρ2 势能的可分离性

使用许多特殊的哈密顿量，可以通过在多个坐标系中分离变量来找到薛定谔方程的解。所涉及的势类包括许多重要的例子，包括各向同性谐振子和库仑势。乘法可分离哈密顿量表现出许多有趣的特征，包括其束缚态谱中的“偶然”简并性，以及通常总是闭合的经典束缚态轨道。我们研究了另一个势，对于它的薛定谔方程在圆柱和抛物线坐标中都是可分的：一个与 $z$ 无关的 $V\propto 1/\rho^{2}=1/(x^{2}+y^{ 2})$ 三个维度。在这个势能中所有持久的、束缚的经典轨道都关闭了，因为所有其他负能量的轨道都落在 $\rho=0$ 的中心。