With the use of the general covariant matrix 10-dimensional Petiau -- Duffin -- Kemmer formalism in cylindrical coordinates and tetrad there are constructed exact solutions of the quantum-mechanical equation for a particle with spin 1 in presence of an external homogeneous magnetic field. There are separated three linearly independent types of solutions; in each case the formula for energy levels has been found.
With the use of the general covariant matrix 10-dimensional Petiau -- Duffin -- Kemmer formalism in cylindrical coordinates and tetrad there are constructed exact solutions of the quantum-mechanical equation for a particle with spin 1 in presence of an external homogeneous magnetic field. There are separated three linearly independent types of solutions; in each case the formula for energy levels has been found.
With the use of the general covariant matrix 10-dimensional Petiau-Duffin-Kemmer formalism in cylindrical coordinates exact solutions of the quantum-mechanical equation for a particle with spin 1 in the presence of an external homogeneous magnetic field are constructed. Three linearly independent types of solutions are separated; in each case the formula for the energy levels has been found.Within similar technique for the quantum-mechanical equation for a particle with spin 1 and additional intrinsic electromagnetic characteristics - polarizability, exact solutions are found in the presence of an external homogeneous magnetic field.
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 ].
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.
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 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.
We formulate and solve an effective spin-one quantum Rabi model motivated by the physical spin-one sector of the ten-component Duffin--Kemmer--Petiau (DKP) theory. After the constrained DKP components are separated from the three physical spin projections, the internal sector is represented as a qutrit. Under a single-mode and magnetic-dipole approximation, its coupling to a quantized electromagnetic field has the spin-one Rabi form g(a+a†)Sx. We derive the regular spectrum of this model in the Bargmann representation. The exact parity symmetry reduces the three-component problem to independent scalar three-term recurrences, from which parity-resolved spectral functions are obtained as continued fractions. The pole energies generated by eliminating the outer spin components are treated separately, allowing exceptional finite-polynomial states to be identified without confusing them with regular zeros of the continued fractions. Numerical comparison with direct Fock-space diagonalization confirms the regular spectral construction to better than 5×10−13 for a representative parameter set. The result establishes an exact spectral framework for a qutrit generalization of the quantum Rabi model and states explicitly the assumptions required for its connection with spin-one DKP dynamics.