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We revisit the quantum-mechanical two-dimensional harmonic oscillator with an electric field confined to a circular box of impenetrable walls. In order to obtain the energy spectrum we resort to the Rayleigh-Ritz method with polynomial and Gaussian basis sets. We compare present results with those derived recently by other authors. We discuss the limits of large and small box radius and also do some calculations with perturbation theory.

In this short note we resort to the well known Hellmann-Feynman theorem to prove that some non-relativistic Hamiltonian operators support an infinite number of bound states.

We argue that several potentials proposed recently for the analysis of the vibrational-rotational spectra of diatomic molecules and their thermodynamic properties exhibit a flaw. One can easily show that the parameters $D_e $ and $r_e$ in those potentials are not the dissociation energy and equilibrium bond length, respectively, as the proposers believe. We show how to overcome the mistake in a simple and quite general way.

We overview the main equations of the Rayleigh-Ritz variational method and discuss their connection with the problem of simultaneous diagonalization of two Hermitian matrices.

We test the analytical expressions for the first two eigenvalues of the harmonic oscillator with a Gaussian perturbation proposed recently. Our numerical eigenvalues show that those expressions are valid in an interval of the coupling parameter that is greater than the one estimated by the authors. We also calculate critical values of the coupling parameter and several exceptional points in the complex plane.

Computational models are an essential tool for the design, characterization, and discovery of novel materials. Hard computational tasks in materials science stretch the limits of existing high-performance supercomputing centers, consuming much of their simulation, analysis, and data resources. Quantum computing, on the other hand, is an emerging technology with the potential to accelerate many of the computational tasks needed for materials science. In order to do that, the quantum technology must interact with conventional high-performance computing in several ways: approximate results validation, identification of hard problems, and synergies in quantum-centric supercomputing. In this paper, we provide a perspective on how quantum-centric supercomputing can help address critical computational problems in materials science, the challenges to face in order to solve representative use cases, and new suggested directions., Comment: 65 pages, 15 figures; comments welcome

We analyse recent contradictory results and conclusions about the spectrum of the screened Coulomb potential $V(r)=-r^{-1}e^{-C/r}$. The well known Hellmann-Feynman theorem shows that all the bound states of the Coulomb potential ($C=0$) remain bounded as $C$ increases. We derive a simple approximate analytical expression for the eigenvalues for sufficiently small values of the screening parameter $C$ and an approximate asymptotic expression for the asymptotic behaviour of the s-state eigenvalues when $C\rightarrow \infty $. Present results are expected to resolve the discrepancy about the spectrum of the quantum-mechanical model just mentioned.

We use an electric-dipole laser-driven transition to precisely measure the cyclotron-frequency ratios of the pairs $^{42}$Ca$^+$-$^{40}$Ca$^+$, $^{44}$Ca$^+$-$^{40}$Ca$^+$ and $^{48}$Ca$^+$-$^{40}$Ca$^+$ in a 7-tesla Penning trap. A single laser-cooled ($T\approx 1$~mK) ion serves, together with photon-counting and/or photon-imaging units, as a radiofrequency detector covering a broad-band frequency spectrum, in the present case from kHz to a few MHz. Such detectors ($^{40,42,44,48}$Ca$^{\scriptsize{+}}$) allow measuring extremely small forces, with measured normalized sensitivities down to $7.4(3.5)$ yN$/\sqrt{\text{Hz}}$ and $24.9(9.9)$ yN$/\sqrt{\text{Hz}}$ in the MHz and kHz regime, respectively. The direct determination of the ions' amplitudes makes a cyclotron-frequency measurement process more robust against inhomogeneities of the magnetic field and/or deviations of the electric quadrupole field due to mechanical imperfections of the trap., Comment: 6 pages, 3 figures

We compare the well known Rayleigh-Ritz variational method (RRVM) with a recently proposed approach based on supersymmetric quantum mechanics and the Gram-Schmidt orthogonalization method (SSQMGS). We apply both procedures to a particular class of double-well harmonic oscillators that had been conveniently chosen for the application of the latter approach. The RRVM eigenvalues converge smoothly from above providing much more accurate results with less computational effort. Present results show that the unproved SSQMGS upper bounds do not hold.

We analyse a recent application of the Cornell potential with spin-spin interaction to the mass spectra of quarkonia and show that the authors have in fact used the Kratzer-Fues potential. They inadvertently converted one potential into the other by means of an invalid transformation.

We discuss a systematic construction of dimensionless quantum-mechanical equations. The process reduces the number of independent model parameters to a minimum and, at the same time, provides the natural units of length, energy, etc. in a clear, straightforward way. We compare this systematic procedure with the widely adopted one that consists of setting $\hbar=1$. As illustrative examples, we choose some simple one-dimensional models proposed recently for the study of localized states in inhomogeneous media.

We show that the effective secular equation proposed several years ago is suitable for estimating the location of the exceptional points of eigenvalue equations. As an illustrative example we choose the well known Mathieu

We study a general one-mode non-Hermitian quadratic Hamiltonian that does not exhibit $\mathcal{PT}$-symmetry. By means of an algebraic method we determine the conditions for the existence of real eigenvalues as well as the location of the exceptional points. We also put forward an algebraic alternative to the generalized Bogoliubov transformation that enables one to convert the quadratic operator into a simpler form in terms of the original creation and annihilation operators. We carry out a similar analysis of a two-mode oscillator that consists of two identical one-mode oscillators coupled by a quadratic term.

We show that a method proposed recently, based on the characteristic polynomial of an effective Hamiltonian, had been developed several years earlier by other authors in a clearer and more general way. We outline both implementations of the approach and compare them by means of the calculation of the exceptional point closest to origin for a toy model.

We discuss results obtained recently for a quantum-mechanical model given by a neutral particle with a magnetic quadrupole moment in a radial electric field and a scalar potential proportional to the radial variable in cylindrical coordinates that also includes the noninertial effects of a rotating reference frame. We show that the conjectured allowed values of the cyclotron frequency are a mere artifact of the truncation of the power series used to solve the radial eigenvalue equation. Our analysis proves that the analytical expression for the eigenvalues are far from correct.

In this Paper we analyze a model proposed recently with the purpose of studying the effects of rotation on the interaction of a point charge with a uniform magnetic field in an elastic medium with a spiral dislocation. In particular we focus on the approximation proposed by the authors that consists of changing the left boundary condition in order to obtain analytical results. We show that this approximation leads to quantitative and qualitative errors, the most relevant one being a wrong prediction of the level spacing.

We show that the straightforward application of the discriminant to some physical problems may yield a trivial useless result. If the symmetry of the model matrix does not change with variations of the model parameter the discriminant may vanish for all values of the parameter due to degeneracy. We illustrate this problem by means of a simple $6\times6$ matrix representation of an Hermitian Hamiltonian operator.

We discuss some simple H\"uckel-like matrix representations of non-Hermitian operators with antiunitary symmetries that include generalized $\mathcal{PT}$ (parity transformation followed by time-reversal) symmetry. One of them exhibits extremely broken antiunitary symmetry (complex eigenvalues for all nontrivial values of the model parameter) because of the degeneracy of the operator in the Hermitian limit. These examples illustrate the effect of point-group symmetry on the spectrum of the non-Hermitian operators. We construct the necessary unitary matrices by means of simple graphical representations of the non-Hermitian operators.

We analyze a recent application of two Adomian-based approximate methods to the Chen-Lee-Liu equation. We prove that the outcome of these approaches is merely the Taylor expansion of the solution about $t=0$ and, consequently, they are valid only for sufficiently small values of the time variable $t$.

We give a simple proof of the well known fact that the approximate eigenvalues provided by the Rayleigh-Ritz variational method are increasingly accurate upper bounds to the exact ones. To this end, we resort to the variational principle, mentioned in most textbooks on quantum chemistry, and to a well known set of projection operators. We think that present approach may be suitable for an advanced course on quantum mechanics or quantum chemistry.

We study the behavior of a quantum particle trapped in a confining potential in one dimension under multiple sudden changes of velocity and/or acceleration. We develop the appropriate formalism to deal with such situation and we use it to calculate the probability of transition for simple problems such as the particle in an infinite box and the simple harmonic oscillator. For the infinite box of length $L$ under two and three sudden changes of velocity, where the initial and final velocity vanish, we find that the system undergoes quantum revivals for $\Delta t = \tau_0 \equiv \frac{4mL^2}{\pi\hbar}$, regardless of other parameters ($\Delta t$ is the time elapsed between the first and last change of velocity). For the simple harmonic oscillator we find that the states obtained by suddenly changing (one change) the velocity and/or the acceleration of the potential, for a particle initially in an eigenstate of the static potential, are {\sl coherent} states. For multiple changes of acceleration or velocity we find that the quantum expectation value of the Hamiltonian is remarkably close (possibly identical) to the corresponding classical expectation values. Finally, the probability of transition for a particle in an accelerating harmonic oscillator (no sudden changes) calculated with our formalism agrees with the formula derived long time ago by Ludwig and recently modified by Dodonov~\cite{Dodonov21}, but with a different expression for the dimensionless parameter $\gamma$. Our probability agrees with the one of ref.~\cite{Dodonov21} for $\gamma \ll 1$ but is not periodic in time (it decays monotonously), contrary to the result derived in ref.~\cite{Dodonov21}., Comment: 29 pages, 10 figures

We apply the Frobenius (power-series) method to some simple exactly-solvable and conditionally-solvable quantum-mechanical models with supposed physical interest. We show that the supposedly exact solutions to radial eigenvalue equations derived in recent papers are not correct because they do not satisfy some well-known theorems. We also discuss the origin of the mistake by means of the approach indicated above.

We discuss a recently proposed approach termed Q-BOR-FDTD method and develop its main equations in a clearer and more rigorous way. We show that it is unsuitable for the calculation of the eigenfunctions in the case of degenerate states and propose an improvement to overcome such limitation.

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.

Two-ion crystals constitute a platform for investigations of quantum nature that can be extended to any ion species or charged particle provided one of the ions in the crystal can be directly laser-cooled and manipulated with laser radiation. This paper presents the formation of two-ion crystals for quantum metrology in a 7-tesla open-ring Penning trap. $^{40}$Ca$^+$ ions are produced either internally by photoionization or externally in a (Paul-trap) source, transported through the strong magnetic field gradient of the superconducting solenoid, and captured in-flight with a mean kinetic energy of a few electronvolts with respect to the minimum of the Penning-trap potential well. Laser cooling of the two-ion crystal in a strong magnetic field towards reaching the quantum regime is also presented with particular emphasis on the cooling of the radial modes.

We obtain the eigenvalues and eigenfunctions of the singular harmonic oscillator $V(x)=\alpha/(2x^2)+x^2/2$ by means of the simple and straightforward Frobenius (power-series) method. From the behaviour of the eigenfunctions at origin we derive two branches for the eigenvalues for negative values of $\alpha$. We discuss the well known fact that there are square-integrable solutions only for some allowed discrete values of the energy.

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., Comment: arXiv admin note: substantial text overlap with arXiv:2011.07015

We show that some non-Hermitian Hamiltonian operators with tridiagonal matrix representation may be quasi Hermitian or similar to Hermitian operators. In the class of Hamiltonian operators discussed here the transformation is given by a Hermitian, positive-definite, diagonal operator. We show that there is an important difference between open boundary conditions and periodic ones. We illustrate the theoretical results by means of two simple, widely used, models.

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 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 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 obtain accurate eigenvalues of the one-dimensional Schr\"odinger equation with a Hamiltonian of the form $H_{g}=H+g\delta (x)$, where $\delta (x)$ 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 Schr\"{o}dinger 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., Comment: arXiv admin note: text overlap with arXiv:2010.04997

We analyze the solution of the Schr\"odinger 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.

The coherent coupling between a quartz electro-mechanical resonator at room temperature and trapped ions in a 7-tesla Penning trap has been demonstrated for the first time. The signals arising from the coupling remain for integration times in the orders of seconds. From the measurements carried out, we demonstrate that the coupling allows detecting the reduced-cyclotron frequency ($\nu_+$) within times below 10~ms and with an improved resolution compared to conventional electronic detection schemes. A resolving power $\nu_+/\Delta \nu_+=2.4\times10^{7}$ has been reached in single measurements. In this publication we present the first results, emphasizing the novel features of the quartz resonator as fast non-destructive ion-trap detector together with different ways to analyze the data and considering aspects like precision, resolution and sensitivity., Comment: 19 pages, 10 figures

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., Comment: arXiv admin note: text overlap with arXiv:2010.04997

We obtain eigenvalues and eigenfunctions of the Schr\"{o}dinger 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.

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.

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., Comment: arXiv admin note: text overlap with arXiv:2009.07039, arXiv:2008.03376

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., Comment: arXiv admin note: text overlap with arXiv:2008.13604

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.

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.

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.

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., Comment: arXiv admin note: substantial text overlap with arXiv:2008.03376