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arXiv:1109.0379

Exact solutions for a quantum-mechanical particle with spin 1 and additional intrinsic characteristics in a homogeneous magnetic field

V. V. Kisel, Elena M. Ovsiyuk, Viktor M. Red'kov, N. G. Tokarevskaya

Published Sep 2, 2011• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1109.0379

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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 ].
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.
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.