Separation of variables in a spheroconical coordinate system and the Schr\" odinger equation for a case of noncentral forces

I. Lukach ; Ya. A. Smorodinskii

Teoretičeskaâ i matematičeskaâ fizika, Tome 14 (1973) no. 2, pp. 170-179

Résumé

A study is made of the separation of variables in a spheroconical coordinate system associated with the existence of an elliptic coordinate system on a three-dimensional sphere. In the class of admissible potentials, interest attaches to a potential of the form $qr^{-4}[3(\boldsymbol\alpha\mathbf r) (\boldsymbol\beta\mathbf r)-(\boldsymbol{\alpha\beta})\mathbf r^2]$, where $\boldsymbol\alpha$ and $\boldsymbol\beta$ are two arbitrary unit vectors. The angular part of this potential has the form of a noncentral interaction similar to the angular part of the interaction between two magnetic dipoles. After the angular part has been reduced to principal axes, the solution of the Schrödinger equation with such a potential leads to the Lamé wave equation. Solutions are found in the first order of perturbation theory, and a study is made of the splitting of the energy levels of a centrally symmetric field when a noncentral potential of this kind is presented. In particular, the energy level splitting is calculated in the presence of such a potential in the case of the Coulomb potential and a potential with a quadratic dependence on the radius.