Spin 1 particle in the magnetic monopole potential, nonrelativistic approximation. Minkowski and Lobachevski spaces

Abstract

The spin 1 particle is treated in the presence of the Dirac magnetic monopole in the Minkowski and Lobachevsky spaces. Separating the variables in the frame of the matrix 10-component Duffin-Kemer-Petiau approach (wave equation) and making a nonrelativistic approximation in the corresponding radial equations, a system of three coupled second order linear differential equations is derived for each type of geometry. For the Minkowski space, the nonrelativistic equations are disconnected using a linear transformation, which makes the mixing matrix diagonal. The resultant three unconnected equations involve three routs of a cubic algebraic equation as parameters. The approach allows extension to the case of additional external spherically symmetric fields. The Coulomb and oscillator potentials are considered and for each of these cases three series of energy spectra are derived. A special attention is given to the states with minimum value of the total angular momentum. In the case of the curved background of the Lobachevsky geometry, the mentioned linear transformation does not disconnect the nonrelativistic equations in the presence of the monopole. Nevertheless, we derive the solution of the problem in the case of minimum total angular momentum. In this case, we additionally involve a Coulomb or oscillator field. Finally, considering the case without the monopole field, we show that for both Coulomb and oscillator potentials the problem is reduced to a system of three differential equations involving a hypergeometric and two general Heun equations. Imposing on the parameters of the latter equations a specific requirement, reasonable from the physical standpoint, we derive the corresponding energy spectra.