The non-relativistic problem for a spin 3/2 particle in magnetic field, and tetrad gauge transformations

In the present paper, we will focus on the non-relativistic problem for a spin 3/2 particle in magnetic field, applying cylindrical coordinates and two tetrads: Cartesian and cylindrical. Here appear six different presentations for 4-component wave functions: three Cartesian ones L cart Ψ cart ,Ψ cart and provided by using the relevant gauge transformation L cyl = S (φ) L cart three different presentations in cylindrical tetrad L cyl Ψ cyl ,Ψ cyl First, we specify the non-relativistic equation for a spin 3/2 particle in magnetic field for Cartesian tetrad in bases with non-diagonal and diagonal matrix of the third spin projection. Solutions of two types are found: the first one is associated with the operator of the orbital angular momentum; the second solution relates to the third projection of the total angular momentum. Equations arising here are solved in terms of the confluent hypergeometric functions, and the corresponding energy spectra are found. The gauge transformation is introduced which relates two tetrads: Cartesian and cylindrical; it permits us to transform the system of equations in polar coordinate from Cartesian tetrad to cylindrical one. The rules for gauge transformations of diagonalized operators of the total angular and orbital momentums are found.