Spin 1/2 particle with two masses in magnetic field

Abstract

In the present paper, the generalized equation for spin 1/2 particle with two mass states is investigated in presence of an external uniform magnetic field. After the separation of variables in cylindric coordinates, the problem reduces to a system of 8 first order differential equations, from where follows the system of related four second order differential equations. After the diagonalization of the mixing term, the separate equations for four functions are derived, in which the spectral parameters coincide with the roots of the 4-th order polynomial. The solutions are constructed in terms of confluent hypergeometric functions, and the analytical formulas for the two series of energy spectrum are found in explicit form as solutions of 4-th order algebraic equations; however these prove to be cumbersome and useless for our purposes. The numerical study of the energy levels is performed depending the parameter γ, determining the mass values, on the magnitude of magnetic field and magnetic and main quantum numbers: E = E1,2(γ, B; m, n). In particular, it is shown that the physical energy spectrum for a two mass fermion differs significantly from the energy spectrum of an ordinary Dirac particle.