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arXiv · 2006.06793

Separability of the Planar $1/\rho^{2}$ Potential In Multiple Coordinate Systems

Richard DeCosta·Brett Altschul
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Abstract

With a number of special Hamiltonians, solutions of the Schr\"{o}dinger 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 Schr\"{o}dinger 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 Schr\"{o}dinger 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.

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Richard DeCosta, Brett Altschul. 2020-06-11. Separability of the Planar $1/\rho^{2}$ Potential In Multiple Coordinate Systems. https://doi.org/10.3390/sym12081312

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