One-dimensional hydrogen atom

Abstract

The theory of the one-dimensional (1D) hydrogen atom was initiated by a 1952 paper but, after more than 60 years, it remains a topic of debate and controversy. The aim here is a critique of the current status of the theory and its relation to relevant experiments. A 1959 solution of the Schrödinger equation by the use of a cut-off at x = a to remove the singularity at the origin in the 1/| x | form of the potential is clarified and a mistaken approximation is identified. The singular atom is not found in the real world but the theory with cut-off has been applied successfully to a range of four practical three-dimensional systems confined towards one dimension, particularly their observed large increases in ground state binding energy. The true 1D atom is in principle restored when the short distance a tends to zero but it is sometimes claimed that the solutions obtained by the limiting procedure differ from those obtained by solution of the basic Schrödinger equation without any cut-off in the potential. The treatment of the singularity by a limiting procedure for applications to practical systems is endorsed.

A Distributional Approach for the One-Dimensional Hydrogen Atom

“…After giving a complete characterization of the subfamily of all parity invariant (attractive) Coulomb interactions, we conducted a systematic study of the bound states for all possible interactions in this subfamily. A summary of the results obtained is presented in Thus, our results, obtained from a mathematically welldefined treatment of the 1D Hydrogen atom, clarify the highly controversial issues concerning the boundedness of the ground state and the parity as well as the degenerescence of the bound states (see [6] and references therein). Finally, this work demonstrates that the method developed in Calçada et al [30] for contact interactions can be generalized for long-range interactions having point singularities (in fact, it makes essential use of the results derived in Calçada et al [30]).…”

“…It is only for particular values of the parameters that the interaction distribution is invariant, as seen below. 6 It is important to notice that choosing h + = +∞ (h − = +∞) implies ψ 0 + = 0 [ψ 0 − = 0]. The choice h ± ∈ R ∪ {−∞} (instead of h ± ∈ R ∪ {+∞}) in these expressions does not change the b.c., resulting in the same interaction.…”

“…and it has been a source of considerable interest since the paper by Loudon [1], who investigated this problem due to its relevance for the physics of excitons in strong magnetic fields. The 1D Coulomb potential finds applications in several fields, such as the quasi one-dimensional hydrogen atom in astrophysical systems with very strong magnetic fields [2], in quantum wires and carbon nanotubes (see, e.g., [3][4][5] and references therein), etc.-for a recent review of the applications of the 1D Coulomb interaction see Loudon [6]. Despite its many applications and deceptive simplicity, the 1D hydrogen atom has been a source of great controversy in the literature.…”

“…Several treatments, including Loudon's [1], regularize the potential by introducing a small cutoff a via V(x) = γ /(|x| + a), and taking the limit a → 0 at the end. This regularization procedure has led to conflicting results in which concerns the degeneracy or not of the spectrum, the existence of even wave functions and whether the associated spectrum is continuous or discrete, and the stability of the model (unboundedness of the ground state energy from below, in the a → 0 limit) [1,6,9,10]. This is, perhaps, unsurprising, since regularization procedures need to be employed with great care and often need additional input, such as symmetry, to yield meaningful results, as is well-known in quantum field theory [11] or in the study of chemical indices [12,13]-see [14,15] for rigorous treatments of the regularized 1D Coulomb potential, and [16] for a general treatment of regularized Sturm-Liouville operators.…”

“…Thus, in section 3 we show that requiring parity invariance further reduces the allowed interactions to either a two-parameter interaction (permeable origin) or an one-parameter interaction (impermeable origin) [30]. We then present, in section 4, a systematic study of the bound states for the subfamily of all parity invariant interactions, addressing questions that are at the origin of the controversies about the 1D hydrogen atom in the literature, such as the multiplicities of bound state energies, the parity of the corresponding eigenfunctions and the boundedness of the ground state (see [6,7,26] and references therein)-as far as we know, such a systematic investigation, including all possible parity invariant b.c., is missing in the literature and it may help to clarify the controversies mentioned above. The results are discussed in section 5 and two appendices, on essentials of distribution theory and the solutions of the Whittaker equation, are included for convenience.…”