Quark tunneling in a one-dimensional nuclear model

A general result for the eigenenergies of the one-dimensional Dirac equation for an arbitrary array of delta-function potentials of either Lorentz scalar or vector type is developed. This result is then used to discuss a finite one-dimensional model of the nucleus from which the energy bands of the constituent quarks may be obtained. The resulting energy eigenvalues are compared with those obtained previously via a similar model based on a Klein-Gordon equation. In contrast to this alternative treatment a system consisting of massless quarks confined to a regular periodic one-dimensional nucleus is found to be unbound. In the near ultra-relativistic limit binding to first order in the mass is found, thus conforming unexpectedly with the nonrelativistic limit. The possible surface states are examined and it is found that the so-called Dirac surface states are not expected for potentials of either Lorentz character. The existence and effect of quark surface states is also discussed in terms of our nuclear model. Finally the scattering coefficients for a relativistic particle in an arbitrary array of delta-function potentials are determined and their implications with regard to quark tunneling in nuclei are discussed.