Particle with Spin 1 in Spherical Riemann Space: Pauli Approximation in Spherically Symmetric Potentials

Abstract

A particle with spin 1 in the spherical Riemann space S3 is treated in presence of the Dirac magnetic monopole in non-relativistic approximation. First, in the relativistic Duffin–Kemmer–Peteau equation the separation of the variables is performed with the use of the Wigner D-functions. Thus there arise three quantum numbers (E, j, m): the energy, the square and the third projection of the generalized total angular momentum. In the radial system of equations, transition to the non-relativistic approximation is performed, the problem is reduced to the system of three interrelated differential equations of second order. The resulting system is very complex, complete analysis is possible only for a special case of minimal value of the quantum number jmin, when additionally external spherically symmetric electric fields can be taken into account. The cases of Coulomb and oscillator potential are studied in detail, and the exact wave functions and energy spectra of the particle have been constructed. In absence of an external monopole potential, the non-relativistic spin 1 particle in spherical space is studied in presence of external Coulomb potential. The differential equations have been solved in terms of Heun functions, exact energy spectra have been found.