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Proceedings of the Komi Science Centre of the Ural Division of the Russian Academy of Sciences

St¨uckelberg particle in external magnetic field. Nonrelativistic approximation. Exact solutions

Elena M. Ovsiyuk, A. Safronov, Alina V. Ivashkevich, O. A. Semenyuk

Published Dec 20, 2022• 1 citation
Publisher

Abstract

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.

DOI
10.19110/1994-5655-2022-5-79-88

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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.
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, 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 ].
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.
The generalized Duffin–Kemmer equation for a spin 1 particle with the anomalous magnetic moment in the external uniform magnetic field is investigated. The separation of variables in the wave equation is performed on the basis of projective operator techniques. The problem is reduced to a system of differential equations for three independent functions that have been solved in terms of the confluent hypergeometric functions. Three series of the energy levels are found. To assign them the physical sense at all values of the main quantum number n = 0,1, 2, , special restrictions on anomalous magnetic moment values must be imposed – they are formulated in explicit form. Otherwise, only some part of the energy levels corresponds to the bound states. The neutral spin 1 particle is considered as well. In this case, no bound states exist in the systems. The main qualitative manifestation of the anomalous magnetic moment is the space scaling of the arguments of the wave functions in comparison with a particle without such a moment.
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 paper, we examine the non-relativistic approximation in the relativistic system of equations in Cartesian coordinates for 16-component wave functions with transformation properties of the vector-bispinor under the Lorentz group. When performing the non-relativistic approximation, for separating large and small components in the complete wave function we apply the method of projective operators. Accordingly, the complete wave function is presented as a sum of three parts: the large part depends on 6 variables, and the small ones depend on 14 variables. We have found two linear constraints on large components and two constraints on the small ones. After performing the procedure of the non-relativistic approximation we have derived 6 equations with a needed non-relativistic structure, which include only 4 large components. It is proved that only 4 equations are independent, so we have arrived at the generalized Pauli-like equation for the 4-component wave function. The analysis of transformation properties of the non-relativistic wave function permits us to generalize the structure of the derived equation to an arbitrary curved 3-space.