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Journal of Physics A Mathematical and General

Exact solutions of the Dirac equation with surface delta interactions

Francisco Domínguez-Adame

Published Jun 7, 1990• 31 citations
Publisher

Abstract

The Dirac equation with vector plus scalar surface delta interactions supported by a sphere is exactly solved for all partial waves and parities. Unlike the relativistic one-dimensional delta potentials, surface delta interactions with finite values of the vector and scalar coupling constants require a minimum strength to bind particles. Strong scalar couplings confine relativistic particles at high energies inside the sphere, while the confinement is no longer possible for strong vector potentials. It is also found that the potential causes no effects on the scattering phase shift in the zero-range limit. Therefore, the well known Dirac-Kronig-Penney model in one dimension cannot be generalised to three-dimensional crystal lattices.

DOI
10.1088/0305-4470/23/11/023

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The Dirac equation with vector plus scalar surface delta interactions supported by a sphere is exactly solved for all partial waves and parities. Unlike the relativistic one-dimensional delta potentials, surface delta interactions with finite values of the vector and scalar coupling constants require a minimum strength to bind particles. Strong scalar couplings confine relativistic particles at high energies inside the sphere, while the confinement is no longer possible for strong vector potentials. It is also found that the potential causes no effects on the scattering phase shift in the zero-range limit. Therefore, the well known Dirac-Kronig-Penney model in one dimension cannot be generalised to three-dimensional crystal lattices.
This paper presents several cases in which the effects of the addition of a delta function potential on bound states can be computed exactly. In the case of the one dimensional Schrödinger equation, a Green’s function technique is used to compute an exact implicit expression that gives the effect of the one−dimensional delta function potential on the bound states eigenvalues. It is also shown that the weak coupling limit of the implicit expression agrees with the perturbative treatment of the delta function. It is pointed out that this technique cannot be applied to obtain an exact treatment of the delta function potential δ3(r) in the three−dimensional Schrödinger equation. Instead, an alternate definition of the delta function in three dimensions as a limit of a sequence of square well potentials is used. Unlike the one dimensional case, the exact effect on the eigenvalue does not agree in the limit of weak coupling with the perturbative results. This indicates that the three−dimensional delta function potential should not be used to approximate strongly coupled short range potentials for which perturbation theory is inadequate.
We investigate the planar Dirac equation with the most general time-independent contact (singular) potential supported on a circumference. Taking advantage of the radial symmetry, the problem is effectively reduced to a one-dimensional one (the radial), and the contact potential is addressed in a mathematically rigorous way using a distributional approach that was originally developed to treat point interactions in one dimension, providing a physical interpretation for the interaction parameters. The most general contact interaction for this system is obtained in terms of four physical parameters: the strengths of a scalar and the three components of a singular Lorentz vector potential supported on the circumference. We then investigate the bound and scattering solutions for several choices of the physical parameters, and analyze the confinement properties of the corresponding potentials.
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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.
We discuss the quantum Hamiltonian HG̳ describing a δ-sphere interaction introduced in [J. Math. Phys 30, 2275 (1989)] and formally given by HG̳=HD+G̳δ(|x|−R), where HD is the Dirac Hamiltonian and G̳ is a real 4×4 matrix defined by G̳=(0A B0). We obtain a series of new results for this model, in particular the resolvent equation, the spectral properties, the nonrelativistic limit and the various quantities related to the scattering theory. These results are generalized to the case of an asymmetric δ-sphere interaction and a δ-sphere plus Coulomb interaction, respectively.