Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Green function for an axially symmetric potential field: a path integral evaluation in polar coordinates

M. Victoria Carpio-Bernido

Published Jul 7, 1991• 27 citations
Publisher

Abstract

An explicit evaluation of the path integral in spherical polar coordinates for the Green function of a particle moving in an axially symmetric potential field is presented. A closed form of the Green function is obtained from which the energy eigenvalues and the normalized eigenfunctions are obtained as the poles and the residues at the poles, respectively.

DOI
10.1088/0305-4470/24/13/016

Related papers

Path integral treatment of a noncentral electric potential

Ali Ghoumaid, Farid Benamira, Larbi Guechi +1 more

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.

Open Physics•Oct 29, 2012•1 citation

Dirac particle in Aharonov–Bohm–Coulomb field: Path integral treatment

Abdeldjalil Merdaci, N. Boudiaf, L. Chétouani

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.

Modern Physics Letters A•Dec 21, 2021•0 citations

Path integral formalism for the free Dirac propagator in spherical coordinates

Sreya Banerjee, Z. Harman

The relativistic Green's function of a free spin-1/2 fermion is derived using the Feynman path integral formalism in spherical coordinates. The Green's function is reduced to an exactly solvable path integral by an appropriate coordinate transformation. The result is given in terms of spherical Bessel functions and spherical spinors, and agrees with previous solutions of the problem.

arXiv:2309.13688•Sep 24, 2023•0 citations

Coulomb Potentials by Path Integration

Ch. Grosche

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.

Fortschritte der Physik/Progress of Physics•Jan 1, 1992•40 citations

Exact Path-Integral Solution of the Dirac-Coulomb Problem

M. A. Kayed, Akira Inomata

An exact path-integral solution of the Dirac equation with a Coulomb potential is found. The energy-dependent Green's function of the second-order Dirac-Coulomb equation, expressed as a polar-coordinate path integral, is reduced to the exactly solvable path integral for an isotropic harmonic oscillator by a coordinate transformation combined with a local time rescaling. The Green's function of the linear Dirac equation is evaluated with the help of the Biedenharn transformation. The energy spectrum for the bound states is also obtained.

Physical Review Letters•Jul 9, 1984•27 citations

Approximate path integral solution for a Dirac particle in a deformed Hulthén potential

A Kadja, Farid Benamira, Larbi Guechi

The problem of a Dirac particle moving in a deformed Hulthén potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green’s function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green’s function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green’s function. A Dirac particle in the standard Hulthén potential ( q = 1) and a Dirac hydrogen-like ion ( q = 1 and a → ∞) are considered as particular cases.

arXiv:1911.10332•Physics of Particles and Nuclei Letters•May 1, 2017•1 citation
An explicit evaluation of the path integral in spherical polar coordinates for the Green function of a particle moving in an axially symmetric potential field is presented. A closed form of the Green function is obtained from which the energy eigenvalues and the normalized eigenfunctions are obtained as the poles and the residues at the poles, respectively.
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.
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.
An exact path-integral solution of the Dirac equation with a Coulomb potential is found. The energy-dependent Green's function of the second-order Dirac-Coulomb equation, expressed as a polar-coordinate path integral, is reduced to the exactly solvable path integral for an isotropic harmonic oscillator by a coordinate transformation combined with a local time rescaling. The Green's function of the linear Dirac equation is evaluated with the help of the Biedenharn transformation. The energy spectrum for the bound states is also obtained.
The problem of a Dirac particle moving in a deformed Hulthén potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green’s function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green’s function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green’s function. A Dirac particle in the standard Hulthén potential ( q = 1) and a Dirac hydrogen-like ion ( q = 1 and a → ∞) are considered as particular cases.