Spin–orbit coupling for the motion of a particle in a ring‐shaped potential

Abstract

Abstract As is known, the Schrödinger equation for a particle in the ring‐shaped potential V ( r , v ) = ησ 2 (2 a 0 / r − a / r 2 sin 2 v )ε 0 , defined in the whole space, has been solved exactly. Here the eigenfunctions are represented in a form which is advantageous for concrete evaluations. The spin–orbit interaction energy E LS in quasirelativistic approximation is determined analytically, for the first time with a nonspherically symmetric potential. The influence of spin–orbit interaction on the eigenvalues of the spin‐free problem and on the selection rules for electrical dipole transitions are investigated as well as the dependence of E LS on the position and depth of the potential minimum. The model can be useful for investigations of axial symmetric subjects like the benzene molecule or related problems and may be easily extended to a many‐electron theory.