Spherical solutions of the wave equation for a spin 3/2 particle

Abstract

The wave equation for a spin 3/2 particle, described by 16-component vector-bispinor, is investigated in spherical coordinates. In the frame of the Pauli–Fierz approach, the complete equation is split into the main equation and two additional constraints, algebraic and differential. The solutions are constructed, on which 4 operators are diagonalized: energy, square and third projection of the total angular momentum, and spatial reflection, these correspond to quantum numbers { }. ε, , , jmP After separating the variables, we have derived the radial system of 8 first-order equations and 4 additional constraints. Solutions of the radial equations are constructed as linear combinations of the Bessel functions. With the use of the known properties of the Bessel functions, the system of differential equations is transformed to the form of purely algebraic equations with respect to three quantities aa a 123 , , . Its solutions may be chosen in various ways by solving the simple linear equation Aa A a Aa 11 2 2 3 3 ++= 0, where the coefficients Ai are expressed trough the quantum numbers ε, j. Two most simple and symmetric solutions have been chosen. Thus, at fixed quantum numbers { } ε, , , jmP there exists double-degeneration of the quantum states.