Path integral treatment of a noncentral electric potential

Abstract

Abstract:We present a rigorous path integral treatment of a dynamical system in the axially symmetric potentialIt is shown that the Green's function can be calculated in spherical coordinate system for V (θ) = . As an illustration, we have chosen the example of a spherical harmonic oscillator and also the Coulomb potential for the radial dependence of this noncentral potential. The ring-shaped oscillator and the Hartmann ring-shaped potential are considered as particular cases. When α = β = γ = 0, the discrete energy spectrum, the normalized wave function of the spherical oscillator and the Coulomb potential of a hydrogen-like ion, for a state of orbital quantum number ≥ 0, are recovered.

The quantum characteristics of a class of complicated double ring-shaped non-central potential

“…The Dirac case was also treated in [25,26]. However, as pointed out in [27], the results in [24] are not coincident. This is because when α = β = γ = 0, the quantum number l = 1 + 2ν + |m|, (ν, |m| ∈ N) does not allow for the s (l = 0) state.…”

“…This is because when α = β = γ = 0, the quantum number l = 1 + 2ν + |m|, (ν, |m| ∈ N) does not allow for the s (l = 0) state. Nevertheless, the path integral method was used in [27] to study this problem again; the replacement of θ → θ/2 also deserves doubt. This is because the problem was studied only in the interval θ ∈ (0, π/2).…”