Cox’s Particle in Magnetic and Electric Fields on the Background of Hyperbolic Lobachevsky Geometry

Abstract

Generalized non-relativistic Schr¨odinger equation for Cox’s not point-like scalar particle with intrinsic structure has been solved exactly in presence of external uniform magnetic and electric fields in the case of Euclidean space. Extension of these problems to the case of the hyperbolic Lobachevsky 3-space is examined. Complete separations of the variables in the system of special cylindric coordinates in this curved model has performed for cases of magnetic and electric fields. In presence of the magnetic field, the quantum problem in radial variable has been solved exactly, wave functions and corresponding energy level have been found; the quantum motion in z-direction is described by 1-dimensional Schr¨odinger-like equation in an effective potential which turns out to be too difficult for analytical treatment. In the presence of electric field on the background of curved model, the situation is similar: radial equation is solved exactly in hypergeometric functions, an equation in z-variable can be be treated only qualitatively because its complexity.