Geometrization for a Quantum-Mechanical Problem of a Vector Particle in an External Coulomb Field

Abstract

The quantum mechanical problem for a relativistic spin 1 particle is studied. With the use of the space reflection operator, the radial system of ten equations is splitted into independent subsystems, consisting of 4 and 6 equations, respectively. The last one reduces to a system of 4 linked first order differential equations for the complex radial functions fi(r) i = 1 4. We investigate this system by using the tools of the Jacobi stability theory, namely, the Kosambi–Cartan–Chen (KCC) theory. It has been shown that the second KCC-invariant is a function of the radial coordinate r, and it does not depend on xi and yi = dxidr , i = 1 8.Due to this, the remaining 3 KCC-invariants identically vanish. In accordance with the general theory, a pencil of geodesic curves from the point r0 converges (or diverges) if the real parts of all eigenvalues of the 2-nd KCC-invariant P ij are negative (or positive). We determine the expressions for the matrix P ij(r) for r 0 and for r , and examine the asymptotic behavior of the eigenvalue problem P = . The established behavior of eigenvalues correlates with the existence of two solutions which may be associated with the bound states of a particle in a Coulomb field. We further describe a method which permits to examine projections of the whole set of solutions onto different 2-planes of the 4-space. In each case, such a projection consists of two branches, which are characterized by different 2-nd order differential equations.