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

Carrillo-Bernal, M. A.; Núñez‐Yépez, H. N.; Salas‐Brito, A. L.; Solis, Didier A.

doi:10.1103/physreve.91.027301

Cited by 3 publications

(3 citation statements)

References 16 publications

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“…This soft-Coulomb interaction has been used in a wide variety of applications requiring a 1d potential [8][9][10][11][12][13][14][15][16]. It has many attractive features including the avoidance of the singularity at zero separation while retaining a Rydberg series of excitations [17][18][19]. Even when not considering a strong electromagnetic field, the soft-Coulomb interaction has become the choice potential for 1d model systems.…”

Section: Introductionmentioning

confidence: 99%

One-dimensional mimicking of electronic structure: The case for exponentials

et al. 2015

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An exponential interaction is constructed so that one-dimensional atoms and chains of atoms mimic the general behavior of their three-dimensional counterparts. Relative to the more commonly used soft-Coulomb interaction, the exponential greatly diminishes the computational time needed for calculating highly accurate quantities with the density matrix renormalization group. This is due to the use of a small matrix product operator and to exponentially vanishing tails. Furthermore, its more rapid decay closely mimics the screened Coulomb interaction in three dimensions. Choosing parameters to best match earlier calculations, we report results for the one dimensional hydrogen atom, uniform gas, and small atoms and molecules both exactly and in the local density approximation.

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Section: Introductionmentioning

confidence: 99%

One-dimensional mimicking of electronic structure: The case for exponentials

et al. 2015

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“…All of the even-parity states are removed again by Gomes & Zimerman [43] but reinstated by Moss [44]. Debate on the solutions of the singular Schrödinger equation (1.1) persists, for example, in [45][46][47] and references therein.…”

Section: Criticisms Of the Basic Theorymentioning

confidence: 99%

One-dimensional hydrogen atom

Loudon¹

2016

Proc. R. Soc. A.

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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.

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“…Many other approaches, such as the use of generalized Laplace transform [17], Fourier transform [18], superpotentials [8], among others [19][20][21][22][23], have also been employed in the solution of the 1D hydrogen atom. Still, these either have not addressed or have not been able to unequivocally resolve all the issues above mentioned.…”

Section: Introductionmentioning

confidence: 99%

A Distributional Approach for the One-Dimensional Hydrogen Atom

et al. 2019

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We consider the one-dimensional Hydrogen atom, with the Coulomb interaction V(x) = γ |x| (γ < 0), and use Schwartz's theory of distributions to address the non-integrable singularity at the origin. This singularity renders the interaction term V(x)ψ(x) in the Schrödinger's equation, where ψ(x) is the wave function, an ill-defined product in the ordinary sense. We replace this ill-defined product by a well-defined interaction distribution, S[ψ, V](x), and by imposing that it should satisfy some fundamental mathematical and physical requirements, we show that this distribution is defined up to a 4-parameter family of contact interactions, in agreement with the method of self-adjoint extensions. By requiring that the interaction distribution be invariant under parity, we further restrict the 4-parameter family of interactions to the subfamily of all the parity invariant Coulomb interactions. Finally, we present a systematic study of the bound states within this subfamily, addressing the frequently debated issues of the multiplicity and parity of the bound states, and the boundedness of the ground state energy.

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

Cited by 3 publications

References 16 publications

One-dimensional mimicking of electronic structure: The case for exponentials

One-dimensional mimicking of electronic structure: The case for exponentials

One-dimensional hydrogen atom

A Distributional Approach for the One-Dimensional Hydrogen Atom

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