On describing bound states for a spin 1 particle in the external Coulomb field

Abstract

The system of 10 radial equations, derived from the Duffin–Kemmer–Petiau equation for a spin 1 particle in the external Coulomb field, is studied. With the use of the space reflection operator, the whole system is split to independent subsystems, consisting of 4 and 6 equations, respectively. The most simple subsystem of 4 equations is solved in terms of hypergeometric functions, which gives the known energy spectrum. Also, the solutions and energy spectrum are found for the minimal value of the total angular momentum, j = 0. The second subsystem is expected to give the description of the other two series of bound states. With the use of the Lorentz generalized condition in presence of the Coulomb field, we prove that one of 6 radial function turns to be equal to zero. This simplifies the explicit form of the system of 6 equations, which contains only 5 unknown functions. Combining this system, we derive a new separated of 2-nd order system of differential equations for three radial functions. In particular, one of the equations turns out to be a rather simple one, and may be recognized as a confluent Heun equation. A series of bound states is constructed in terms of the so called transcendental confluent Heun functions, which provides us with solutions for the second class of bound states, with corresponding formula for energy levels. The subsystem of 6 equations, with no use of additional constraints due to the Lorentz condition, after excluding two non-differential relations reduces to the system of 1-st order differential equations for 4 functions fi, i = 1, 2, 3, 4. We derive the explicit form of a corresponding of 4-th order equation for each function. Among them, there are equations with two substantially different sets of singular points: 3 regular (or 2) and 2 irregular of rank 2. Any of these functions may be considered as a main one, and all remaining functions may be found in explicit form, in terms of the main one. From the four independent solutions of each 4-th order equation, only two solutions may be referred to series of bound states.