Spin 1 particle in magnetic monopole potential, nonrelativistic approximation, Minkowski and Lobachevsky space models

O.V. Veko, K.V. Kazmerchuk, V.V. Kisel, E.M. Ovsiyuk, V.M. Red'kov

National Academy of Sciences of Belarus, Belarus

Spin 1 particle is treated in presence of the Dirac magnetic monopole. The separation of the variables is done in Duffin-Kemer-Petiau equation. In the radial equations, the nonrelativistic approximation has been performed, there arise system of three 2-nd order differential equations. These equations can be disconnected with the use of special linear transformation making the mixing matrix diagonal. As result, there arise three separated equations which contain routs $A_ {k}$ of a cubic algebraic equation. The algorithm permits extension to the presence of external spherically symmetrical fields, the cases of Coulomb and oscillator ones are treated in detail. For each case, there are found three series of energy spectra $\epsilon = \epsilon(A_{k},j, n)$. Analysis is extended to the case of Lobachevsky geometry. After performing transition to the non-relativistic description. we derive a system of second order interrelated equations, which cannot be disconnected in presence of the monopole. Progress is possible only in presence of pure Coulomb and pure oscillator fields, both the problems reduced to Heun equation. With the use of special requirement,reasonable from physical standpoint energy spectra are obtained.