Abstract
We consider the quantum mechanical problem of the motion of a spinless charged relativistic particle with mass M, described by the Klein-Fock-Gordon equation with equal scalar Sr→ and vector Vr→ Coulomb plus ring-shaped potentials. It is shown that the system under consideration has both a discrete at E<Mc2 and a continuous at E>Mc2 energy spectra. We find the analytical expressions for the corresponding complete wave functions. A dynamical symmetry group SU1,1 for the radial wave equation of motion is constructed. The algebra of generators of this group makes it possible to find energy spectra in a purely algebraic way. It is also shown that relativistic expressions for wave functions, energy spectra, and group generators in the limit c⟶∞ go over into the corresponding expressions for the nonrelativistic problem.
Highlights
Nonrelativistic Schrödinger and relativistic Dirac, KleinFock-Gordon (KFG), and finite-difference equations describe the systems in the nuclear physics, elementary particle physics, and atomic and molecular physics [1,2,3,4,5,6]
Other significant noncentral potentials of the form (1) were considered by Hautot [16] in the framework of the nonrelativistic quantum mechanics to study the problem of the motion of a charged particle
We present here the exact solutions of the KFG equation with equal scalar and vector potentials for the sum Coulomb and ring-shaped potentials of the type (1)
Summary
Nonrelativistic Schrödinger and relativistic Dirac, KleinFock-Gordon (KFG), and finite-difference equations describe the systems in the nuclear physics, elementary particle physics, and atomic and molecular physics [1,2,3,4,5,6]. Other significant noncentral potentials of the form (1) were considered by Hautot [16] in the framework of the nonrelativistic quantum mechanics to study the problem of the motion of a charged particle He considered the two- and three-dimensional harmonic oscillator potentials and the Coulomb potential to which terms of the type f ðθÞ/r2 were added and found such functions f ðθÞ that enabled him to solve exactly the corresponding Schrödinger equations. One needs to study the noncentral potentials rather than the central ones for getting better results in molecular structures and interactions Examples to these situations include the use of the ring-shaped potentials in quantum chemistry to describe the ring-shaped organic molecules and in nuclear physics to investigate the interaction between the deformed nucleus and the spin-orbit coupling for a motion of the particle in the potential fields.
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