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Chinese Journal of Physics

Path Integral for the Coulomb Problem in the Feshbach-Villars Formalism

Tahar Boudjedaa, L. Chétouani, M. Merad

Published Dec 1, 2000• 5 citations
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Abstract

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.

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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.
In this paper we have set up a path integral formalism for Feshbach-Villars equation by using the fermionic Schwinger model for Pauli matrices which describe an isocharge symmetry. This choice is made in analogy with spin model and the coherent state representation is then used. We have also given a general method of treating the problem of vanishing scalar potential by reducing it the to non-relativistic case and then, via Foldy-Wouthuysen canonical transformation, an explicit solution is constructed. The free case and constant magnetic field interaction are explicitly exposed. In each cases the propagators are evaluated and the energy spectrum and the corresponding wave functions are deduced. P.A.C.S.03.65 Ca.Formalism P.A.C.S.03.65 Db.Functional analytical methods P.A.C.S.03.65 Pm.Relativistic wave equations P.A.C.S.03.65 Ge.Solutions of wave equations: bound states.
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.
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 investigate a central coordinate system within the framework of the Wigner–Dunkl deformation of quantum mechanics in d spatial dimensions, focusing on the dynamics of a relativistic spinless particle described by the Klein–Gordon equation. The analysis is carried out using Feynman’s path integral formalism in hyperspherical coordinates, where the Lagrangian includes an effective potential depending explicitly only on radial variables. This approach allows for the exact separation of variables in d-dimensional Dunkl space, enabling an analytical evaluation of the path integral. Particular emphasis is placed on the attractive Coulomb-type potential, for which the exact propagator is obtained in closed form. As a result, we derive the energy spectrum and corresponding wave functions of the Dunkl–Klein–Gordon system, expressed in terms of special functions that reflect the modified symmetries induced by the Dunkl operators.
Using the coherent states representation and the Schwinger boson model, the propagator of a spinning particle in interaction with an arbitrary magnetic field is derived in the framework of the path integral formalism. To show the relevance of the formalism to some applications, the Green function of spinning particle s=1/2 in special configuration of magnetic field is calculated. The transition amplitude and Berry’s phase for an arbitrary spin and time dependent magnetic field are deduced. The case of half integer spin is also considered.