Herein, a spin 1 particle with anomalous magnetic moment in an external Coulomb field is studied. We start with the relativistic tensor system of the Proca type in Cartesian coordinates. In these equations the Γ parameter is present related to an additional characteristic of the particle. In the case of an external magnetic field, it is interpreted as an anomalous magnetic moment. In the presence of an external electric field, additional interaction terms are presented as well; moreover, the terms of the first and second orders in parameter Γ appear. The case of an external Coulomb field is considered in detail. In the nonrelativistic approximation a Pauli type equation is obtained. In the nonrelativistic equation the separation of the variables with the use of spherical vectors is realized. One separate 2-nd order differential equation is found, in which additional interaction terms are missing. Besides, we derive systems of two coupled 2-nd order equations wherein linear and quadratic in parameter Γ interaction terms are presented. Previously, another approach was developed for analyzing the vector particle with anomalous magnetic moment. It was based on the use of tetrad formalism and separation of the variables in the Duffin – Kemmer equation with the help of the Wigner function. The nonrelativistic approximation was performed directly in the system of radial equations. Besides, previously formal Frobenius type solutions for an arising 4-th order differential equation were constructed; however, physically interpretable energy spectra were not found. We have proved that the radial equations derived by different methods are the same up to a simple liner transformation over two radial functions. In this paper, we have obtained a simpler 4-th order equation, the construction of Frobenius solutions becomes technically easier, but physical energy spectra are not found either.

The tetrad-based generalized complex formalism by Majorana–Oppenheimer is applied to examine an electromagnetic field in oscillating de Sitter Universe in nonstatic spherically symmetric coordinates. With the help of Wigner D-functions we separate the angular (Θ, φ) -dependence in the complex vector field E j ( x ) + iB j ( x ) from the ( t , r )-dependence. After that, the system of differential equations in (t, r) variables is solved exactly. Relations between the complex 3-vector Majorana–Oppenheimer formalism and the 10-component Duffin–Kemmer–Petiau approach have been examined. On this basis, electromagnetic waves of magnetic and electric types have been constructed in the both formalisms. In the Duffin–Kemmer–Petiau formalism, the class of gradient-type solutions is constructed in Coulomb and Lorentz gauges.

The system of nonrelativistic Pauli equations for a spin 1 particle is solved in the case of the Riemann space of constant positive curvature. The system of three interrelated radial equations is divided into two subsystems with the use of a space reflection operator: one and two equations, respectively. The first is solved in hypergeometric functions straightforwardly. The second subsystem gives two 4-order ordinary differential equations; they are solved with the use of the factorization method: energy spectra and wave functions 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.

For expanding de Sitter space-time, a spin 1 particle is investigated in the non-relativistic Pauli approximation. After separation of the variables in the relativistic Duffin–Kemmer–Petiau equation, the procedure of non-relativistic approach is performed for the system of 10 equations in the variables (t, r). As a result, the problem reduces to three second-order related differential equations. Requirement of diagonalization of the parity operator allows the system to be split into (1 + 2) subsystems. The fourth-order equations obtained are solved with the help of the factorization method, which permits the problem to be reduced to the analysis of second-order equations. In this way, the Pauli equation for a spin 1 particle in the expanding De Sitter universe is solved exactly: three series of states and the relevant rules of quantization of the spectral parameter are obtained.

On the basis of the method of separation of variables, the complete set of exact solutions of the Dirac equation in the non-static coordinates of the de Sitter space is constructed. In the separation of variables, the formalism of the Wigner D -functions is used. The square and the third projection of the total angular momentum, as well as the space reflection operators are diagonalized on the solutions. Equations for the radial variable lead to a discrete spectrum of the separation constant. The asymptotic properties of the solutions for radial ant time variables are investigated.

The Dirac equation for spin 1/2 particle with anomalous magnetic moment is solved in presence of the external uniform magnetic field. After separation of the variables, the problem is reduced to a 4-order ordinary differential equation, which is solved exactly with the use of the factorization method. A generalized formulas for Landau energy levels are found. Solutions are expressed in terms of confluent hypergeometric functions.

In the 30-component first-order wave equation (Fedorov, 1951) for a massive spin 2 particle, a non-relativistic approximation is performed. The quantum-mechanical equation of Pauli type for a spin 2 particle in the presence of an external electromagnetic field is derived. The non-relativistic wave function is a symmetric irreducible 2-rank tensor with five independent components.

Квантово-механічна частка зі спіном 1 досліджується в поле магнітного заряду в нерелятивістському наближенні. Після розділення змінних завдання зводиться до системи зачіпляються диференціальних рівнянь другого порядку для трьох радіальних функцій. За допомогою спеціального лінійного перетворення система радіальних рівнянь розділяється і завдання зводиться до дослідження трьох рівнянь однакової структури, кожне з яких містить в якості параметра свій корінь кубічного рівняння, що виникає при вирішенні завдання приведення до діагонального вигляду змішаної матриці в системі рівнянь. Аналіз допускає узагальнення на випадки присутності сферично-симетричних потенційних полів, зокрема, кулонівського і осцілляторного

Spin 1 particle is treated in presence of magnetic monopole in nonrelativistic approximation. After separation of the variables the problem is reduced to the system of three interrelated equations, which can be disconnected with the use of special linear transformation making the mixing matrix diagonal. As result, there arise three separated differential equations which contain roots (i 1 2 3) i A    of a cubic algebraic equation as parameters. The algorithm permits extension to the presence of external spherically symmetrical fields, in particular, Coulomb and oscillator ones. Keywords: magnetic monopole, Duffin–Kemmer equation, the non-relativistic approximation.