The one dimensional Hydrogen atom revisited

The one dimensional Schroedinger hydrogen atom is an interesting mathematical and physical problem to study bound states, eigenfunctions and quantum degeneracy issues. This 1D physical system gave rise to some intriguing controversy over more than four decades. Presently, still no definite consensus seems to have been reached. We reanalyzed this apparently controversial problem, approaching it from a Fourier transform representation method combined with some fundamental (basic) ideas found in self-adjoint extensions of symmetric operators. In disagreement with some previous claims, we found that the complete Balmer energy spectrum is obtained together with an odd parity set of eigenfunctions. Closed form solutions in both coordinate and momentum spaces were obtained. No twofold degeneracy was observed as predicted by the degeneracy theorem in one dimension, though it does not necessarily have to hold for potentials with singularities. No ground state with infinite energy exists since the corresponding eigenfunction does not satisfy the Schroedinger equation at the origin.Comment: Accepted for publication in the Canadian Journal of Physics, July 28th, 200

One-dimensional hydrogen atom: A singular potential in quantum mechanics

A generalized Laplace transform approach is developed to study the eigenvalue problem of the one-dimensional singular potential V = -e2/\x\. Matching of solutions at the origin that has been a matter of much controversy is, thereby, made redundant. A discrete and non-degenerate bound-state spectrum results. Existing arguments in the literature that advocate (a) a continuous spectrum, (b) a degeneracy of energy levels as a result of a hidden O(2) symmetry, (c) an infinite negative energy state and (d) an impenetrable barrier at the origin are found to be untenable. It is argued that a judicious use of generalized functions, coupled with some classical considerations, enables the conventional method of solving the problem to recover precisely the same results which are shown to be in accord with an accurate semiclassical analysis of the problem