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Proceedings of the National Academy of Sciences of Belarus Physics and Mathematics Series

Eigenvalues of the generalized helicity operator for spin 3/2 particle in the presence of the magnetic field and the projective operators method

A. V. Ivashkevich, Viktor M. Red'kov

Published Jan 6, 2026• 1 citation
Publisher

Abstract

The eigenvalue problem for generalized helicity operator for a spin 3/2 particle in presence of the uniform magnetic field is solved. After separating the variables in the basis of cylindrical coordinates (r, ϕ, z) and the tetrad, the system of 16 first-order differential equations in the variable r is derived. This system is studied with the use of the method of projective operators, constructed with the use of the third projection of the spin for the particle. In accordance with thе method by Fedorov – Gronskiy, all 16 variables may be expressed in terms of only 4 distinguished functions, which are constructed in terms of confluent hypergeometric functions. Further the problem reduces to studying the linear algebraic homogeneous system for 16 algebraic variables. In the end, we derive algebraic equations of the second and the fourth order, their roots determine the possible eigenvalues of the helicity operator.

DOI
10.29235/1561-2430-2025-61-4-307-319

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The eigenvalue problem for generalized helicity operator for a spin 3/2 particle in presence of the uniform magnetic field is solved. After separating the variables in the basis of cylindrical coordinates (r, ϕ, z) and the tetrad, the system of 16 first-order differential equations in the variable r is derived. This system is studied with the use of the method of projective operators, constructed with the use of the third projection of the spin for the particle. In accordance with thе method by Fedorov – Gronskiy, all 16 variables may be expressed in terms of only 4 distinguished functions, which are constructed in terms of confluent hypergeometric functions. Further the problem reduces to studying the linear algebraic homogeneous system for 16 algebraic variables. In the end, we derive algebraic equations of the second and the fourth order, their roots determine the possible eigenvalues of the helicity operator.
Equation for a spin 1/2 neutral particle possessing three additional characteristics is examined in uniform magnetic and electric fields. Solutions with cylindrical symmetry are searched. On solutions, operators of the energy and the third projection of the total angular momentum are diagonalized. After separating the variables, a system of 4 linked differential equations in partial derivatives with respect to cylindrical coordinates (r,z) is derived. With the use of a special algebraization method, the rrr-dependence of the searched solutions is fixed through the confluent hypergeometric functions. Then a system of 4 linked differential equations in z-coordinate is derived; which is solved as well with the use of algebraization method. The constructed wave functions depend substantially on three additional characteristics. Analytical results are studied numerically. tissue.
In this chapter, within the general method of Gel'fand–Yaglom, starting with an extended set of representations of the Lorentz group, we have constructed a relativistic generalized system of the first-order equations for a spin 1 particle with two additional characteristics, anomalous magnetic moment and polarizability. We have taken into account the presence of external electromagnetic fields. After eliminating the accessory variables of the complete wave function, we derive the minimal 10-component form of equations, the last includes two additional interaction terms which are interpreted as related to anomalous magnetic and moment and polarizability of the particle. This approach is extended to space-time models with pseudo-Riemannian structure, within the tetrad method. We specify the basic equations to the cylindrical coordinates, and also in presence of external uniform magnetic field. After separating the variables, we get the system of ten first-order-differential equations in polar coordinate. To resolve this system, we apply the method of Fedorov–Gronskiy; and the complete 10-component wave function is decomposed into the sum of three projective constituents, where dependence of each component on the polar coordinate is determined by only one function, https://www.w3.org/1998/Math/MathML" display="inline"> F i ( r ) , i = 1 , 2 , 3 . We construct them in terms of confluent hypergeometric equations; at this a quantization rule for some spectral parameter exists. Additionally arises an algebraic homogeneous system of ten equations, which completely determines the structure of 10-component solutions. By vanishing the determinant of the linear system, we derive a cubic algebraic equation with respect to energy parameter ϵ 2 . Its solutions are found in analytical form, and studied numerically. For different values of parameters, anomalous magnetic moment and polarizability, we have found three series of physically interpretable energy levels. For more details see [ 111 ]–[ 119 ].
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.
In this chapter, 16-component system of equations describing the Dirac–Kähler particle in presence of the external uniform magnetic field has been studied. This equation describes a multi-spin boson field equivalent to the scalar, pseudoscalar, vector, pseudovector, and anti-symmetric tensor. On the searched solutions, we diagonalize operators of the energy, third projection of the total angular momentum, and the third projection of the linear momentum. After separating the variable, we derive the system of 16 first-order differential equations in polar coordinate. To resolve this system, we apply the method of Fedorov–Gronskiy. We have found five independent solutions. The number 5 for independent solutions correlates with five different tensors entering the complete wave function for Dirac–Kähler particle [ 67 ].
The St¨uckelberg equation for a particle with two spin states, S = 1 and S = 0, is studied in the presence of an external uniform magnetic field. In relativistic case, the particle is described by an 11-component wave function. On the solutions of the equation, the operators of energy, the third projection of the total angular momentum, and the third projection of the linear momentum along the direction of the magnetic field are diagonalized. After separation of variables, we derive a system for 11 functions depending on one variable. We perform the nonrelativistic approximation in this system. For this we apply the known method of deriving nonrelativistic equations from relativistic ones, which is based on projective operators related to the matrix Γ0 of the relativistic equation. The nonrelativistic wave function turns out to be 4-dimensional. We derive the system for 4 functions. It is solved in terms of confluent hypergeometric functions. There arise three series of energy levels with corresponding solutions. This result agrees with that obtained for the relativistic St¨uckelberg equation.