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Open Physics

Path integral treatment of a noncentral electric potential

Ali Ghoumaid, Farid Benamira, Larbi Guechi, Zohra Khiat

Published Oct 29, 2012• 1 citation

Abstract

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.

DOI: 10.2478/s11534-012-0125-9

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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.

We have studied the path integral solution of a system of particle moving in a certain class of noncentral potential without using the Kustannheimo–Stiefel transformation. The Hamiltonian of the system has been converted to a separable Hamiltonian of Liouville type in parabolic coordinates and has further reduced to a Hamiltonian corresponding to two two-dimensional simple harmonic oscillators. The energy spectrum for this system is calculated analytically. The Hartmann ring-shaped potential and compound Coulomb plus the Aharanov–Bohm potential have also been studied as special cases.

We have studied the path integral solution of a system of particle moving in certain class of non-central potential without using Kustannheimo-Stiefel transformation. The Hamiltonian of the system has been converted to a separable Hamiltonian of Liouville type in parabolic coordinates and has further reduced to a Hamiltonian corresponding to two 2- dimensional simple harmonic oscillators. The energy spectrum for this system is calculated analytically. Hartmann ring-shaped potential and compound Coulomb plus Aharanov- Bohm potential have also been studied as special cases.

Exact Green’s function related to a Dirac particle submitted to the combination of Aharonov–Bohm and Coulomb potentials in [Formula: see text]) coordinate space is analytically calculated via path integral formalism. The Pauli matrices which describe the spin dynamics are replaced by two fermionic oscillators via the Schwinger model. The energy spectrum as well as the corresponding normalized wave functions are extracted following this approach. The interesting properties of the spinors are thus deduced after symmetrization. According to the symmetric form for Green’s function, it is shown that the non-relativistic limit of the Dirac particle is undertaken with much ease.

The path integral for the potential is evaluated in three different coordinate systems, i. e. in cartesian coordinates, in polar coordinates and in parabolic coordinates. The equivalence between the different approaches is shown and in each case the energyspectrum and the wave functions are explicitly calculated. Furthermore we discuss special cases, including the ring-potential, the Coulomb potential with an Aharonov-Bohm solenoid, and the genuine Coulomb problem. We also point out the separability of the path integral formulation of the ring-potential in spheroidal coordinates, and of the Coulomb potential in spheroidal and spheroconical coordinates, respectively.