The one-dimensional hydrogen atom revisited

Abstract

The one dimensional Schrödinger 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 Schrödinger equation at the origin. PACS number(s): 03.65.Ge 03.65.-w *

Phase Space Path Integral Representation for Wigner Function

“…is Bohr radius. It coincides with Bohr radius in three dimension [25]. Left plot of Figure 8 shows dependence of average energy on inverted temperature for SCC potential.…”

“…Sometimes the SCC potential is used in numerical calculations instead of true Coulomb potential to avoid Coulomb divergency [25]. is Bohr radius.…”

Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit

“…Since the beginning of an element can lie near an extremum, in which case h i can be too large, we will enforce a similar requirement for the change in first derivative as well. Following a similar procedure, we get yet another step size from a quadratic equation with coefficients [F (4) …”

“…An alternative option for this method is to combine the above two requirements into a single fourth-order approximation and get h i by selecting the minimum in magnitude of the roots from the resulting cubic equation with coefficients [F (4) …”

“…However, whether the corresponding wave functions possess only even, or only odd, or both even and odd parity, has been a major topic of controversy for many years [4]. Moreover, the existence of an even parity ground state with an energy of −∞ and a δ(x) (Dirac δ) wave function is another topic of disagreement among researchers.…”