On solutions of the Dirac equation for 1D hydrogenic atoms or ions

Abstract Studies of 1D hydrogenic atoms or ions reveal interesting physics and could be relevant to the following experimental situations: (1) hydrogenic atoms in very strong magnetic fields where the electron is effectively confined in a finite-radius cylinder surrounding a field line; (2) an electron constrained to the surface of a nanotube in the field of an ion; (3) excitons in research of semiconductors, superconductivity, and polymers; (4) electron at the surface of liquid helium. In the present theoretical paper, we revisited the 1D Dirac–Coulomb problem. We employed a model where the Coulomb potential is truncated by a constant value at some relatively small but finite distance R from the origin, thus allowing for the finite size of the nucleus of charge Z . We derived analytically the wave functions in the exterior and interior regions. We showed that for nonzero energies, the wave functions in the exterior and interior regions and their derivatives cannot be matched at the boundary, so that the corresponding mathematical solutions are not physically acceptable. For zero energy, we determined the value of R ( Z ), at which the interior and exterior wave functions and their derivatives can be matched at the boundary. We demonstrated that for the electronic hydrogenic atoms or ions, the matching value of R ( Z ) is greater than the corresponding nuclear sizes by at least two orders of magnitude. However, for the muonic hydrogenic atoms or ions, the matching value of R ( Z ) turned out to be approximately equal to the nuclear size R 0 ( Z ) (within the accuracy of the expressions for R 0 ( Z )) for the range of the nuclear charges 14 ≤ Z ≤ 25 ( i.e. , from Si to Mn), the best match of R and R 0 being for Z = 18 (Ar) and Z = 19 (K).