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Modern Physics Letters A

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

Abdeldjalil Merdaci, N. Boudiaf, L. Chétouani

Published Dec 21, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217732321502692

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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.
In this paper, we propose to construct a path integral formalism for a spinless particle in the Feshbach-Villars formalism. The Pauli matrices describing the isocharge symmetry are replaced by the bosonic Schwinger model and the coherent state representation is then used. The radial free case is determined via the Foldy-Wouthuysen canonical transformation and the important case of the Coulomb potential is investigated. The Green functions are evaluated and the energy spectrum and corresponding wave functions are exactly deduced. PACS. 03.65.Db – Functional analytical methods. PACS. 03.65.Pm – Relativistic wave equations. PACS. 03.65.Ge – Solutions of wave equations: bound states. I.
Abstract The Green’s functions related to a Dirac electron and its corresponding antiparticle, i.e., the positron submitted to an exponentially decaying magnetic field in (2 + 1) coordinate space are analytically calculated via path integral formalism. The energy spectra for both discrete and continuous phenomena, as well as the corresponding normalized eigenspinors are extracted following this approach from the known Morse potential propagator. The interesting properties of the spinors are thus deduced after symmetrization of the Green’s functions. Special cases are also considered.
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.
In this paper, we consider Dirac fermion confined in harmonic potential and submitted to a constant magnetic field. The corresponding solutions of the energy spectrum are obtained by using the path integral techniques. For this, we begin by establishing a symmetric global projection, which provides a symmetric form for the Green function. Based on this, we show that it is possible to end up with the propagator of the harmonic oscillator for one charged particle. After some transformations, we derive the normalized wave functions and the eigenvalues in terms of different physical parameters and quantum numbers. By interchanging quantum numbers, we show that our solutions have interesting properties. The density of current and the nonrelativistic limit are analyzed where different conclusions are obtained. Finally, the completeness of the Dirac oscillator eigenfunctions is proved by using the standard properties of the generalized Laguerre polynomials.