62ª Reunião Anual da SBPC

A. Ciências Exatas e da Terra - 3. Física - 5. Física das Partículas Elementares e Campos

DIRAC EQUATION WITH VECTOR AND SCALAR POTENTIALS VIA SUSY QM

Eriverton da Silva Rodrigues 1 Aércio Ferreira de Lima 1 Rafael de Lima Rodrigues 1

1. Departamento de Física, Universidade Federal de Campina Grande / UFCG

INTRODUÇÃO:

We consider a spin 1/2 particle in a potential which is the sum of the Coulomb potential Vc and a Lorentz scalar potential Vs(r). The scalar potential is added to the mass term in the Dirac equation and may be interpreted as an effective position dependent mass. If the scalar potential is assumed to be created by the exchange of mass less scalar mesons, it has the form Vs. The (1+3) and (1+1) dimensional Dirac equations with both scalar-like and vector-like potentials is well known in the literature for a long time. Exact solutions for the bound states in this mixed potential can be obtained by the method of separation of variables and also by the use of the dynamical algebra SO(2, 1).

METODOLOGIA:

In a recent paper the solution of the scattering problem for this potential has been obtained by an analytic method and also by an algebraic method by Vaidya and Silva Souza. In this work we consider an alternative calculation for the energy spectrum and eigenfunctons of the Dirac equation via a Schrodinger-like wave equation, for the vector and scalar potentials introduced by Alhaidari and recently discussed by Chen. In this work, a spin 1/2 relativistic particle described by a general potential in terms of the sum of the Coulomb potential with a Lorentz scalar potential is investigated via generalized ladder operators from supersymmetry (SUSY) shape-invariant Hamiltonians in quantum mechanics. This formalism is applied for the generalized Dirac-Coulomb problem which is an exactly solvable potential in relativistic quantum mechanics.

RESULTADOS:

Our calculation generalizes the results previously obtained for the relativistic Coulomb problem. Instead of a direct generalization of Sukumar's calculation we use the equivalent approach developed by Fukui and Aizawa and also by Balantekin where shape invariance plays an important role. The Coulomb problem in non-relativistic quantum mechanics has been obtained by the use of SUSY shape invariances. The special case of the Dirac-Coulomb potential has been treated recently via SUSY quantum mechanics by one of the authors. This work is organized as follows. From the time independent Dirac equation for a potential which is the sum of the Coulomb potential with a Lorentz scalar potential we construct the radial equations and using shape invariance we deduce ladder operators to build up the energy eigenvalues and eigenfunctions.

CONCLUSÃO:

In relativistic supersymmetric quantum mechanics formalism ladder operators are constructed to obtain the complete set of the energy spectrum and eigenfunctions for a potential which is the sum of the Coulomb potential with a Lorentz scalar potential inversely proportional to r in Dirac equation. These new generalized ladder operators obtained via supersymmetry shape-invariant Hamiltonians in quantum mechanics can be reduced to the ones of the Coulomb potential, which provides us the exact relativistic energy.

Palavras-chave: SUSY, Dirac, relativistic.