arXiv subjects
Publications and source records attributed to Francisco M. Fernández.
We propose an alternative solution to a quantum-mechanical four-particle system in one dimension with two- and three-particle interactions. The solution of the eigenvalue equation in center-of-mass and Jacobi coordinates is considerably simpler than a procedure proposed recently based on spherical coordinates.
We argue that the results proposed recently for the $\mathrm{a}^3Σ_u^+$ electronic state of $^7\mathrm{Li}_2$ do not exhibit any physical utility because the canonical vibrational partition function for an excited electronic state makes no sense according to the principles of statistical mechanics. Such an approach may, in principle, be applied to the ground state of a diatomic molecule. We show that the simpler model based on the harmonic oscillator and rigid rotor enables one to derive a theoretical expression for the equilibrium constant of a simple chemical reaction that agrees remarkably well with experimental data. It is not clear whether the complicated formulas mentioned above are suitable for any comparison with experimental data.
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.
In this paper we show that several authors have derived wrong physical conclusions from a gross misunderstanding of the exact eigenvalues and eigenfunctions of a conditionally-solvable quantum-mechanical model. It consists of an eigenvalue equation with seemingly Coulomb, linear and harmonic terms. Here we compare the results derived by those authors with the actual eigenvalues of the models calculated by means of the Ritz variational method.
We show a parameter-dependent $3\times 3$ non-Hermitian matrix that exhibits both degeneracy and coalescence of eigenvalues at an exceptional point (Hermitian and non-Hermitian degeneracies). This simple non-Hermitian model is suitable for the discussion of those concepts in an undergraduate or graduate course on quantum-mechanics. We also study the symmetry group responsible for the degeneracy.
We show that a non-Hermitian operator with a tridiagonal matrix representation in a finite-dimensional vector space is similar to an Hermitian operator. The required condition is sufficient and simple examples show that it is not necessary. We derive quite general features of the eigenvalues and eigenvectors for a somewhat particular case.
We show that the Adomian decomposition method proposed by Ke et al [ANZIAM J. \textbf{59} (2018) 349] is just the Taylor series approach in disguise. The latter approach is simpler, more straightforward and yields a recurrence relation free from integrals.
We show that the author of a recent paper [arXiv:2008.04929] put forward some false statements about the eigenstates of Hermitian and non-Hermitian systems. We conjecture that one of the non-Hermitian Hamiltonians for a one-dimensional lattice is similar to an Hermitian one and, consequently, exhibits real eigenvalues. Present theoretical analysis of the eigenvalue equation suggests that one of the sets of numerical results in the criticized paper may not be correct.
We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form $H_{g}=H+gδ(x)$, where $δ(x)$ is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction.
We analyze the application of the "tridiagonal representation approach" (TRA) to the Schrödinger equation for some simple, exactly-solvable, quantum-mechanical models. In the case of the Kratzer-Fues potential the mathematical reasoning appears to exhibit a serious flaw that invalidates the result and the expression for the energy does not appear to be correct. We also show that the well known Frobenius method, which resembles the TRA, is far simpler, clearer and more elegant; in addition to give the correct result.
We analyse the solution to the Thomas-Fermi equation discovered by Majorana. We show that the series for the slope at origin enables one to obtain results of accuracy far beyond those provided by available methods. We also estimate the radius of convergence of this series and conjecture that the singularity closest to origin is a square-root branch point.
We analyze recently published results about the Aharonov-Bohm effect for bound states from the interaction of the magnetic quadrupole moment of a neutral particle with axial fields. We show that the eigenvalues obtained by the authors from an arbitrary truncation of the Frobenius power series correspond to more than one model and, consequently, are unsuitable for drawing any sound physical conclusion. Besides, the prediction of allowed oscillator frequencies is a mere consequence of the truncation condition just mentioned.
We analyze the solution of the Schrödinger equation arising in the treatment of a geometric model introduced to explain the origin of the observed shallow levels in semiconductors threaded by a dislocation density. We show (contrary to what the authors claimed) that the model does not support bound states for any chosen set of model parameters. Assuming a fictitious motion in the $x-y$ plane there are bound states provided that $k\neq 0$ and not only for $k>0$ as the authors believed. The truncation condition proposed by the authors yields only one particular energy for a given value of a chosen model parameter and misses all the others (conditionally solvable problem)
We analyze the conclusions of the influence of a Coulomb-type potential on the Klein-Gordon oscillator. We show that the truncation method proposed by the authors do not yield all the eigenvalues of the radial equation but just one of them for a particular value of a model parameter. Besides, the existence of allowed oscillator frequencies that depend on the quantum numbers is an artifact of the truncation method.
We analyze the results obtained from a model consisting of the interaction between a moving electric quadrupole moment and a magnetic field. We argue that there are no bound states contrary to what the author stated. It is shown that the proposed method is unsuitable for obtaining all the bound states and that the existence of allowed cyclotron frequencies is an artifact of the approach.
We draw attention on the fact that the Riccati-Padé method developed some time ago enables the accurate calculation of bound-state eigenvalues as well as of resonances embedded either in the continuum or in the discrete spectrum. We apply the approach to several one-dimensional models that exhibit different kind of spectra. In particular we test a WKB formula for the imaginary part of the resonance in the discrete spectrum of a three-well potential.
We analyze recent results on a Dirac oscillator. We show that the truncation of the Frobenius series does not yield all the eigenvalues and eigenfunctions of the radial equation. For this reason the eigenvalues reported by the authors are useless and the prediction of allowed oscillator frequencies meaningless.
We analyze recent results concerning the hypothesis of a privileged direction in the space-time that is made by considering a background of the Lorentz symmetry violation determined by a fixed spacelike vector field and the analysis of quantum effects of this background on the interaction of a nonrelativistic electron with a nonuniform electric field produced by a uniform electric charge distribution. We show that the conclusions derived by the authors are an artifact of the truncation of the Frobenius series by means of the tree-term recurrence relation for the expansion coefficients. Thus, the existence of allowed angular frequencies stemming from this procedure is meaningless and unphysical.