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arXiv:0803.1551 · Physica Scripta

Bound states of bosons and fermions in a mixed vector–scalar coupling with unequal shapes for the potentials

Luis B Castro, Antonio S de Castro

Published Mar 20, 2008• 10 citations
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Abstract

The Klein–Gordon and the Dirac equations with vector and scalar potentials are investigated under a more general condition, V v + V s =constant. These intrinsically relativistic and isospectral problems are solved in the case of squared hyperbolic potential functions and bound states for either particles or antiparticles are found. The eigenvalues and eigenfuntions are discussed in some detail and the effective Compton wavelength is revealed to be an important physical quantity. It is revealed that a boson is better localized than a fermion when they have the same mass and are subjected to the same potentials.

DOI
10.1088/0031-8949/77/04/045007

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The Klein–Gordon and the Dirac equations with vector and scalar potentials are investigated under a more general condition, V v + V s =constant. These intrinsically relativistic and isospectral problems are solved in the case of squared hyperbolic potential functions and bound states for either particles or antiparticles are found. The eigenvalues and eigenfuntions are discussed in some detail and the effective Compton wavelength is revealed to be an important physical quantity. It is revealed that a boson is better localized than a fermion when they have the same mass and are subjected to the same potentials.
The Klein–Gordon and the Dirac equations with vector and scalar potentials are investigated under a more general condition, V v = V s +constant. These isospectral problems are solved in the case of squared trigonometric potential functions and bound states for either particles or antiparticles are found. The eigenvalues and eigenfunctions are discussed in some detail. It is revealed that a spin-0 particle is better localized than a spin-1/2 particle when they have the same mass and are subjected to the same potentials.
We study the Dirac equation in 3+1 dimensions with a general combination of scalar, vector and tensor interactions with arbitrary strengths, all of them described by central Coulomb potentials acting on a particular plane of motion. For the tensor coupling a constant term is also included, since this gives rise to an effective Coulomb potential, which is necessary for the formation of bound states in a pure tensor coupling configuration. The exact bound-state solutions for this generalized Coulomb problem are computed by exploiting the freedom in choosing the coefficients of the \textit{Ansätze} for the radial functions, which leads to wave functions in terms of generalized Laguerre polynomials. From the quantization condition, the exact energy spectrum is also determined and its dependence on the parameters of the potentials is discussed. We show that similar features of the equations for the problem in the plane and the spherically symmetric problem allow a simple and direct mapping between them, thereby providing the solution to the spherical Coulomb problem. Our results are validated by showing that the solutions correctly encompass several previous solutions available in the literature for particular cases of this problem, for which we further develop the analysis of the parameters. We also derive two new particular cases not yet reported in the literature: the case of breaking of spin and pseudospin symmetries by the addition of a Coulomb plus constant tensor potential and the problem of a scalar plus tensor Coulomb potentials.