An Interdisciplinary Journal

2018, Vol.21, No.4, pp.309 - 325

Geometrization for a Quantum-Mechanical Problem of a Vector Particle in an External Coulomb Field
N. G. Krylova, E. M. Ovsiyuk, V. Balan, and V. M. Red'kov

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 f i (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 x i and y i = dx i /dr, f i (r), 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 PΨ = λ Ψ. 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.

Key words: quantum mechanics, spin 1 particle, Coulomb field, differential geometry, Kosambi- Cartan-Chern invariants

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