The problem of the one-dimensional hydrogen atom has evoked interest because of its relevance to the behavior of hydrogeniclike atoms in the presence of strong magnetic fields and of hydrogenic impurities confined in quantum-well wire structures. The binding energy of the one-dimensional nonrelativistic hydrogen atom has been found to be infinite in its ground state. We have solved the relativistic hydrogen atom problem for the one-dimensional case using the Klein–Gordon equation. We find that the binding energy in the ground state for the one-dimensional relativistic hydrogen atom is finite and is of order of the rest mass energy of the electron. Therefore a relativistic treatment removes the infinite binding energy found for the ground state for the one-dimensional nonrelativistic hydrogen atom.
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Spector et al. (1985) studied this question.