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Francisco M. Fernández

Publications and source records attributed to Francisco M. Fernández.

At least 73 records · Page 4Linked to original sources
arXiv2020

On the calculation of bound-state energies supported by hyperbolic double well potentials

We obtain eigenvalues and eigenfunctions of the Schrödinger equation with a hyperbolic double-well potential. We consider exact polynomial solutions for some particular values of the potential-strength parameter and also numerical energies for arbitrary values of this model parameter. We test the numerical method by means of a suitable exact asymptotic expression for the eigenvalues and also calculate critical values of the strength parameter that are related to the number of bound states supported by the potential.

quant-ph↗
arXiv2020

Comment on: "Solving many-body Schrödinger equations with kinetic energy partition method'', Ann. Phys. 388 (2018) 54-68 by Y-H. Chen and S. D. Chao

We discuss a recent test of the performance of the kinetic energy partition method (KEP) through its application to two separable quantum-mechanical models. We argue that one of the benchmark models is exceedingly simple for testing any realistic approximate method and that almost any reasonable approach yields better results. In the second example our exact benchmark eigenvalues disagree considerably with those chosen by the authors for comparison, which casts doubts on the accuracy of their KEP approach.

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arXiv2020

Gross misinterpretation of a conditionally solvable eigenvalue equation

We solve an eigenvalue equation that appears in several papers about a wide range of physical problems. The Frobenius method leads to a three-term recurrence relation for the coefficients of the power series that, under suitable truncation, yields exact analytical eigenvalues and eigenfunctions for particular values of a model parameter. From these solutions some researchers have derived a variety of predictions like allowed angular frequencies, allowed field intensities and the like. We also solve the eigenvalue equation numerically by means of the variational Rayleigh-Ritz method and compare the resulting eigenvalues with those provided by the truncation condition. In this way we prove that those physical predictions are merely artifacts of the truncation condition.

quant-ph↗
arXiv2020

Comment on: "Harmonic oscillator in an environment with a pointlike defect". Phys. Scr. \textbf{94} ( 2019) 125301

We analyze recent results for a harmonic oscillator in an environment with a pointlike defect. We show that the allowed oscillator frequencies predicted by the authors stem from a misinterpretation of the exact solutions of a conditionally solvable eigenvalue equation. Also the exact eigenvalues derived by those authors are meaningless because they belong to different quantum-mechanical models.

quant-ph↗
arXiv2020

Comment on: "On the effects of the Lorentz symmetry violation yielded by a tensor field on the interaction of a scalar particle and a Coulomb-type field" Ann. Phys. 399 (2018) 117-123

We analyze the eigenvalues and eigenfunctions stemming from a recent study of the interaction of a scalar particle with a Coulomb potential in the presence of a background of the violation of the Lorentz symmetry established by a tensor field. We show, beyond any doubt, that the physical conclusions drawn by the authors from a truncation of a power series, coming from the application of the Frobenius method, are meaningless and nonsensical.

quant-ph↗
arXiv2020

Dimensionless equations in non-relativistic quantum mechanics

We discuss the numerous advantages of using dimensionless equations in non-relativistic quantum mechanics. Dimensionless equations are considerably simpler and reveal the number of relevant parameters in the models. They are less prone to round-off errors when applying numerical methods because all the quantities are of the other of unity. A dimensionless equation facilitates the application of perturbation theory and provides a glimpse of the sort of solution we are going to obtain beforehand.

quant-ph↗
arXiv2020

The rotating harmonic oscillator revisited

We analyze the distribution of the eigenvalues of the quantum-mechanical rotating harmonic oscillator by means of the Frobenius method. A suitable ansatz leads to a three-term recurrence relation for the expansion coefficients. Truncation of the series yields some particular eigenvalues and eigenfunctions in exact analytical form. The former can be organized in such a way that one obtains suitable information about the whole spectrum of the model.

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arXiv2020

Comment on: "On the influence of a Coulomb-like potential induced by the Lorentz symmetry breaking effects on the harmonic oscillator''. Eur. Phys. J. Plus (2012) \textbf{127}: 102

We analyze the calculation of bound states for a nonrelativistic spin-half neutral particle under the influence of a Coulomb-like potential induced by Lorentz symmetry breaking effects. We show that the truncation condition proposed by the authors only provides one energy eigenvalue for a particular model potential and misses all the other bound-state energies. The dependence of the cyclotron frequency on the quantum numbers is a mere artifact of the truncation condition that is by no means necessary for the existence of bound states.

quant-ph↗
arXiv2020

Comment on: "Some quantum aspects of a particle with electric quadrupole moment interacting with an electric field subject to confining potentials". Int. J. Mod. Phys. A 29 (2014) 1450117

We analyze the results obtained from a model consisting of the interaction etween the electric quadrupole moment of a moving particle and an electric field. We argue that the system does not support bound states because the motion along the $z$ axis is unbounded. It is shown that the author obtains a wrong bound-state spectrum for the motion in the $x-y$ plane and that the existence of allowed cyclotron frequencies is an artifact of the approach.

quant-ph↗
arXiv2020

Comment on: "Rashba coupling induced by Lorentz symmetry breaking effects". Ann. Phys. (Berlin) \textbf{526}, 187 (2013)

We analyze the results of a paper on "the arising of a Rashba-like coupling, a Zeeman-like term and a Darwin-like term induced by Lorentz symmetry breaking effects in the non-relativistic quantum dynamics of a spin-1/2 neutral particle interacting with external fields". We show that the authors did not obtain the spectrum of the eigenvalue equation but only one eigenvalue for a specific relationship between model parameters. In particular, the existence of allowed cyclotron frequencies conjectured by the authors is a mere artifact of the truncation condition used to obtain exact solutions to the radial eigenvalue equation.

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arXiv2020

Algebraic treatment of non-Hermitian quadratic Hamiltonians

We generalize a recently proposed algebraic method in order to treat non-Hermitian Hamiltonians. The approach is applied to several quadratic Hamiltonians studied earlier by other authors. Instead of solving the Schrödinger equation we simply obtain the eigenvalues of a suitable matrix representation of the operator. We take into account the existence of unitary and antiunitary symmetries in the quantum-mechanical problem.

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arXiv2020

Comment on: "Interaction of the magnetic quadrupole moment of a non-relativistic particle with an electric field in a rotating frame. Ann. Phys. 412 (2020) 168040''

We analyze a recent treatment of the interaction of a magnetic quadrupole moment with a radial electric field for a non-relativistic particle in a rotating frame and show that the derivation of the equations in the paper is anything but rigorous. The authors presented eigenvalues and eigenfunctions for two sets of quantum numbers as if they belonged to the same physical problem when they are solutions for two different models. In addition to it, the authors failed to comment on the possibility of multiple solutions for every set of quantum numbers.

quant-ph↗
arXiv2020

The truncated Coulomb potential revisited

We apply the Frobenius method to the Schrödinger equation with a truncated Coulomb potential. By means of the tree-term recurrence relation for the expansion coefficients we truncate the series and obtain exact eigenfunctions and eigenvalues. From a judicious arrangement of the exact eigenvalues we derive useful information about the whole spectrum of the problem and can obtain other eigenvalues by simple and straightforward interpolation.

quant-ph↗
arXiv2020

Comment on: "Relativistic quantum dynamics of a charged particle in cosmic string spacetime in the presence of magnetic field and scalar potential''. Eur. Phys. J. C (2012) 72:2051

We analyze the results of a paper on ``Relativistic quantum dynamics of a charged particle in cosmic string spacetime in the presence of magnetic field and scalar potential''. We show that the authors did not obtain the spectrum of the eigenvalue equation but only one eigenvalue for a specific relationship between model parameters. In particular, the existence of allowed cyclotron frequencies conjectured by the authors is a mere artifact of the truncation condition used to obtain exact solutions to the radial eigenvalue equation.

hep-th↗
arXiv2020

On some conditionally solvable quantum-mechanical problems

We analyze two conditionally solvable quantum-mechanical models: a one-dimensional sextic oscillator and a perturbed Coulomb problem. Both lead to a three-term recurrence relation for the expansion coefficients. We show diagrams of the distribution of their exact eigenvalues with the addition of accurate ones from variational calculations. We discuss the symmetry of such distributions. We also comment on the wrong interpretation of the exact eigenvalues and eigenfunctions by some researchers that has led to the prediction of allowed cyclotron frequencies and field intensities.

quant-ph↗
arXiv2020

Comment on "Bidimensional bound states for charged polar nanoparticles''

In a recent paper Castellanos-Jaramillo and Castellanos-Moreno proposed a simple quantum-mechanical model for an electron in the vicinity of an ionized nanostructure with a permanent electric dipole. They chose the interaction of the electron with the charge and the dipole in such a way that the resulting Schrödinger equation is separable into radial and angular parts. In this comment we show that those authors did not solve the angular eigenvalue equation with proper periodic boundary conditions and that they also made a mistake in the elimination of the first derivative in the radial equation. Such errors invalidate their results of the Einstein coefficients for the $\left( GaAs\right)_{3}$ system considered.

quant-ph↗
arXiv2020

Comment on: "Development of the perturbation theory using polynomial solutions" J. Math. Phys. 60, 012103 (2019)

The purpose of this comment is to present the perturbation approach proposed by Maiz J. Math. Phys. 60, 012103 (2019) in a clearer way. The results of our straightforward procedure agree with those obtained by that author except for one case in which we obtain the exact result while he obtained an approximate one. In addition to it, we show that for sufficiently deep double-well potentials the perturbation approach deteriorates considerably.

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