Generalized Helicity Operator for a Spin 2 Particle in the Presence of an External Uniform Magnetic Field

Abstract

The eigenvalue problem for generalized helicity operator for a spin 2 particle in the presence of a uniform magnetic field is solved. After separating variables in the basis of cylindrical tetrad the system of 10 first order differential equations is derived, it is split into two independent subsystems of four and six equations. First, the free particle is studied. The system of four equations is solved straightforwardly in terms of the confluent hypergeometric functions, there are found corresponding eigenvalues and eigenfunctions. Subsystem of six equations leads to one ordinary 4-th order differential equation. Corresponding operator is factorized into the product of two commuting 2-nd order operators, so the problem reduces to solving two differential equations of the 2-nd order. Their solutions are constructed in terms of the Bessel functions. The problem is extended to a presence of an external uniform magnetic field, the method is much the same, the explicit solutions are constructed in terms of the confluent hypergeometric functions. The helicity eigenvalues are described in an implicit form, as solutions of the polynomial equations of 3-rd and 5-th orders.