Quantum Mechanical Scalar Particle with Intrinsic Structure in External Magnetic and Electric Fields: Influence of Geometrical Background

Abstract

Relativistic theory of the Cox’s scalar not point-like particle is developed in the presence of electromagnetic and gravitational fields. This theory is specified in simple geometrical backgrounds: Euclid’s, Lobachevsky’s, and Riemann’s. Wave equations for the Cox’s particle, relativistic and non-relativistic, are solved exactly in the presence of external uniform magnetic and electric fields in the case of Minkowski space. Non-trivial additional structure of the particle modifies the frequency of a quantum oscillator arising e˙ectively in the presence of an external magnetic field. Extension of these problems to the case of hyperbolic Lobachevsky space is examined. In the presence of a magnetic field, the quantum problem in radial variable has been solved exactly; quantum motion in z-direction is described by 1-dimensional Schr¨odinger-like equation in an e˙ective potential which turns out to be too di˚cult for analytical treatment. In the presence of an electric field, the situation is similar. The same analysis has been performed for spherical Riemann space model. General conclusion can be done: the role of large scale structure of the Universe depends greatly on the form of basic equations for a particle, any modification of them lead to new physical phenomena due to non-Euclidean geometry background.