, Volume 11, Issue 1, pp 78-88
Date: 27 Oct 2012

Path integral treatment of a noncentral electric potential

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Abstract

We present a rigorous path integral treatment of a dynamical system in the axially symmetric potential \(V(r,\theta ) = V(r) + \tfrac{1} {{r^2 }}V(\theta ) \) . It is shown that the Green’s function can be calculated in spherical coordinate system for \(V(\theta ) = \frac{{\hbar ^2 }} {{2\mu }}\frac{{\gamma + \beta \sin ^2 \theta + \alpha \sin ^4 \theta }} {{\sin ^2 \theta \cos ^2 \theta }} \) . 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 l ≥ 0, are recovered.