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

Abstract

As is known, the Schrodinger equation for a particle in the ring-shaped potential V(r.8) =( 2 4 r -&r2 sin2t9)co, 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 Ew. 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 EL^. 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.* Dedicated to Professor Dr. Willy Hartner on the occasion of his 75th birthday t u = (cry, uy, u z ) , with u, = Pauli spin matrices.

Relativistic solution in D-dimensions to a spin-zero particle for equal scalar and vector ring-shaped Kratzer potential

“…The study of exact solutions of the nonrelativistic equation for a class of non-central potentials with a vector potential and a non-central scalar potential is of considerable interest in quantum chemistry [31][32][33][34][35][36][37]. In recent years, numerous studies [38][39][40] have been made in analyzing the bound states of an electron in a Coulomb field with simultaneous presence of Aharonov-Bohm (AB) [41] field, and/or a magnetic Dirac monopole [42], and AharonovBohm plus oscillator (ABO) systems.…”

Path integral treatment of a noncentral electric potential

“…This potential has been proposed by Hartmann as a model for the benzene molecule [1][2][3]. The Green's function relative to this potential can be deduced from expressions (45) and (28),…”

Exact solutions of the D-dimensional Schrödinger equation for a ring-shaped pseudoharmonic potential

“…On the other hand, the study of exact solutions of the SE for non-central physical potentials, resulting from adding a potential proportional to ( sin θ) −2 to the exactly solvable vector (central) Coulombic and harmonic parts, is of considerable interest in quantum chemistry, beginning with the pioneering work of Hartmann [37][38][39] and Makarov et al [40] on ring-shaped potentials. In recent years, numerous studies [41][42][43][44][45] have been made in analyzing the bound states of an electron in a Coulomb field with the simultaneous presence of an Aharanov-Bohm (AB) [46] field, and/or a magnetic Dirac monopole [47], and Aharanov-Bohm plus oscillator (ABO) systems.…”