Erratum: Quantum solution for the one-dimensional Coulomb problem [Phys. Rev. A83, 064101 (2011)]

Abstract

In our paper a superselection rule (SSR) was derived as part of a detailed analysis of the one-dimensional (1D) Coulomb problem. Here we set straight some inaccuracies in our arguments that could give rise to objections over the existence of such SSR. We emphasize that the lack of recognition of such SSR lies at the root of the many erroneous claims made about the system. Our paper is devoted to the discussion of the properties of the 1D Coulomb problem. We exhibit that the peculiar properties of its quantum solutions can be traced back to the complete independence between the right and the left sides of the singularity at z = 0 in the 1D Coulomb potential V (z) = −k/|z|, k > 0. We argued that such feature is related to a SSR effectively separating the left states ψ − n (z), which vanish for z > 0, from the right states ψ + n (z), which vanish for z < 0. However, though the SSR exists and the separation operates, the proof given in our paper is flawed. In this erratum we want to set the argument straight. We work in units such that = e = m = 1.As we proved in our paper, the normalized z-space eigenfunctions of the 1D Coulomb problem are [1,2]

Comment on “Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit”

“…at the origin [8,9], So there cannot be particle flow from the right to the left of the origin or vice versa, in agreement with the impenetrable wall that, as GW mention, appears to develop as fi 0. Unfortunately, GW do not take into account all the con sequences of such behavior.…”

“…Based on the form of the probability density (Figs. 3, 4, and 5 of [I]), GW suggest that, as p -» 0, Vq°(x = 0) = 0, which is the condition making the origin impermeable [5,6,[8][9][10]. The boundary condition i/f(0) = 0, at the hard Coulomb limit, implies the vanishing of the quantum flux,…”

“…3, 4, and 5 of [I]), GW suggest that, as p -» 0, Vq°(x = 0) = 0, which is the condition making the origin impermeable [5,6,[8][9][10]. The boundary condition i/f(0) = 0, at the hard Coulomb limit, implies the vanishing of the quantum flux,• ,*3^ , 3r J=l^ 8 7 -* 1 7 = 0 .(3) *=0at the origin [8,9], So there cannot be particle flow from the right to the left of the origin or vice versa, in agreement with the impenetrable wall that, as GW mention, appears to develop as …”

“…Some of the issues they bring about have already been extensively discussed and ultimately solved. Both the supposed negative infinite energy ground state and the parity eigenstates for the problem have been shown to be nonexistent on mathematical grounds using techniques that are very important for both ID systems and their applications [3][4][5][6][7][8][9][10][11][12],…”

Chemistry in one dimension

“…44 Despite its simplicity, this model has been useful for studying the behavior of many physical systems, such as Rydberg atoms in external fields 45,46 or the dynamics of surface-state electrons in liquid helium 47,48 and its potential application to quantum computing. 49,50 Most work since Loudon has focused on one-electron ions 44,[51][52][53][54][55][56][57] and, to the best of our knowledge, no calculation has been reported for larger chemical systems. In part, this can be attributed to the ongoing controversy concerning the mathematical structure of the eigenfunctions.…”