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European Journal of Physics

Comment on ‘Quantum anharmonic oscillator plus delta-function potential: a molecular view of pairing formation and breaking in the coordinate space’

Francisco Marcelo Fernández

Published Jul 9, 2021• 2 citations
Publisher

Abstract

Abstract We analyze a recent discussion of one-dimensional quantum-mechanical models with a perturbation produced by a delta-function potential. We show that the authors failed to derive a correct transcendental equation for the eigenvalues in the case of the Morse oscillator because they omitted the contribution of the continuum spectrum in the basis set.

DOI
10.1088/1361-6404/ac0fb9

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Reply to the comment on ‘Quantum anharmonic oscillator plus delta-function potential: a molecular view of pairing formation and breaking in the coordinate space’

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Abstract In this short paper we clarify and answer some issues addressed by Fernandez in his comment paper. Firstly, regarding the citation issue and secondly the technical issue regarding the analytical result of the quantum anharmonic oscillator with a delta potential.

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Comment on ‘Numerical estimates of the spectrum for anharmonic PT symmetric potentials’

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We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-$x$ plane (Rep. Prog. Phys. {\bf 70} (2007) 947).

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Comment on `Dimensional expansion for the delta-function potential'

R M Cavalcanti

I criticize the claim, made in a recent article (Bender C M and Mead L R 1999 Eur. J. Phys . 20 117-21) that, in order to obtain the correct cross section for the scattering from a two-dimensional delta-function potential, one must perform analytic continuation in the dimension of space.

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Comment on "Numerical estimates of the spectrum for anharmonic PT symmetric potentials"

Amore, Paulo, Francisco Marcelo Fernández

We show that the authors of the commented paper (Bowen et al 2012 Phys. Scr. 85 065005) draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In some of the studied examples, the author missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operators and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-x plane (Bender 2007 Rep. Prog. Phys. 70 947)

•Mar 1, 2013•3 citations
Abstract We analyze a recent discussion of one-dimensional quantum-mechanical models with a perturbation produced by a delta-function potential. We show that the authors failed to derive a correct transcendental equation for the eigenvalues in the case of the Morse oscillator because they omitted the contribution of the continuum spectrum in the basis set.
Abstract We propose an alternative way to describe the pairing formation and breaking via a quantum anharmonic oscillator with a delta-function potential model. Unlike BCS theory, which describes the pairing formation in the momentum space, this model works in the coordinate space and is able to give a molecular view of pairing formation and breaking in the coordinate space. By exploring the dynamical interplay between the intrinsic factor (dissociation energy) and external factor (pairing strength) of this system additional information was gained, including the critical pairing strength and critical scattering length, which might relate to the BCS–BEC crossover phenomena and halo state formation. Although only the energetic aspect of pairing is described by this model, its simplicity and pedagogical steps might help undergraduate students to understand the pairing problem in a simple way.
Abstract The Morse potential is an important problem to examine due to its applications in describing vibrations and bond breaking in molecules. It also shares some properties with the simpler harmonic oscillator, at the same time displaying differences, allowing for an interesting contrast to its well-studied counterpart. The solution of the Morse potential is not usually taught in a quantum mechanics class, since using differential equations makes it very tedious. Here, we illustrate how to solve the Morse potential using the Schrödinger factorization method. This operator method is a powerful tool to find the energy eigenvalues, eigenstates, and wavefunctions without using differential equations in position space, allowing us to solve more problems without requiring a discussion of hypergeometric or confluent hypergeometric functions.
We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-xxx plane (Rep. Prog. Phys. {\bf 70} (2007) 947).
We show that the authors of the commented paper (Bowen et al 2012 Phys. Scr. 85 065005) draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In some of the studied examples, the author missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operators and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-x plane (Bender 2007 Rep. Prog. Phys. 70 947)