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Francisco M. Fernández

Publications and source records attributed to Francisco M. Fernández.

At least 145 records · Page 8Linked to original sources
arXiv2014

Algebraic treatment of $\mathcal{PT}$-symmetric coupled oscillators

The purpose of this paper is the discussion of a pair of coupled linear oscillators that has recently been proposed as a model of a system of two optical resonators. By means of an algebraic approach we show that the frequencies of the classical and quantum-mechanical interpretations of the optical phenomenon are exactly the same. Consequently, if the classical frequencies are real, then the quantum-mechanical eigenvalues are also real.

quant-ph↗
arXiv2014

Accurate calculation of the solutions to the Thomas-Fermi equations

We obtain highly accurate solutions to the Thomas-Fermi equations for atoms and atoms in very strong magnetic fields. We apply the Padé-Hankel method, numerical integration, power series with Padé and Hermite-Padé approximants and Chebyshev polynomials. Both the slope at origin and the location of the right boundary in the magnetic-field case are given with unprecedented accuracy.

quant-ph↗
arXiv2013

Comparison of two semiclassical expansions for a family of PT-symmetric oscillators

We show that a recently developed semiclassical expansion for the eigenvalues of PT-symmetric oscillators of the form $V(x)=(ix)^{2N+1}+bix$ does not agree with an earlier WKB expression for $V(x)=-(ix)^{2N+1}$ the case $b=0$. The reason is due to the choice of different paths in the complex plane for the calculation of the WKB integrals. We compare the Stokes and anti-Stokes lines that apply to each case for the quintic oscillator and derive a general WKB expression that contains the two earlier ones.

quant-ph↗
arXiv2013

Bound states of a charged particle in the field of an electric quadrupole in two dimensions

The Schrödinger equation for a charged particle in the field of a nonrelativistic electric quadrupole in two dimensions is known to be separable in spherical coordinates. We investigate the occurrence of bound states of negative energy and find that the particle can be bound by a quadrupole of any magnitude. This result is remarkably different from the one for a charged particle in the field of a nonrelativistic electric dipole in three dimensions where a minimum value of the dipole strength is necessary for capture. Present results differ from those obtained earlier by other author.

quant-ph↗
arXiv2013

Eigenvalues and eigenfunctions of the anharmonic oscillator $V(x,y)=x^{2}y^{2}$

We obtain sufficiently accurate eigenvalues and eigenfunctions for the anharmonic oscillator with potential $V(x,y)=x^{2}y^{2}$ by means of three different methods. Our results strongly suggest that the spectrum of this oscillator is discrete in agreement with early rigorous mathematical proofs and against a recent statement that cast doubts about it.

quant-ph↗
arXiv2013

$\mathcal{PT}$-symmetric strings

We study both analytically and numerically the spectrum of inhomogeneous strings with $\mathcal{PT}$-symmetric density. We discuss an exactly solvable model of $\mathcal{PT}$-symmetric string which is isospectral to the uniform string; for more general strings, we calculate exactly the sum rules $Z(p) \equiv \sum_{n=1}^\infty 1/E_n^p$, with $p=1,2,\dots$ and find explicit expressions which can be used to obtain bounds on the lowest eigenvalue. A detailed numerical calculation is carried out for two non-solvable models depending on a parameter, obtaining precise estimates of the critical values where pair of real eigenvalues become complex.

math-ph↗
arXiv2013

Local approximation to the critical parameters of quantum wells

We calculate the critical parameters for some simple quantum wells by means of the Riccati-Padé method. The original approach converges reasonably well for nonzero angular-momentum quantum number $l$ but rather too slowly for the s states. We therefore propose a simple modification that yields remarkably accurate results for the latter case. The rate of convergence of both methods increases with $l$ and decreases with the radial quantum number $n$. We compare RPM results with WKB ones for sufficiently large values of $l$. As illustrative examples we choose the one-dimensional and central-field Gaussian wells as well as the Yukawa potential. The application of perturbation theory by means of the RPM to a class of rational potentials yields interesting and baffling unphysical results.

quant-ph↗
arXiv2012

Particle correlation from uncorrelated non Born-Oppenheimer SCF wavefunctions

We analyse a nonadiabatic self-consistent field method by means of an exactly-solvable model. The method is based on nuclear and electronic orbitals that are functions of the cartesian coordinates in the laboratory-fixed frame. The kinetic energy of the center of mass is subtracted from the molecular Hamiltonian operator in the variational process. The results for the simple model are remarkably accurate and show that the integration over the redundant cartesian coordinates leads to couplings among the internal ones.

quant-ph↗
arXiv2012

Comment on: "Analytical approximations for the collapse of an empty spherical bubble"

We analyze the Rayleigh equation for the collapse of an empty bubble and provide an explanation for some recent analytical approximations to the model. We derive the form of the singularity at the second boundary point and discuss the convergence of the approximants. We also give a rigorous proof of the asymptotic behavior of the coefficients of the power series that are the basis for the approximate expressions.

physics.flu-dyn↗
arXiv2012

On the uncertainty relations for an electron in a constant magnetic field

We discuss the uncertainty relation for the azimuthal angle $ϕ$ and the $z$-component of the angular momentum $L_z$. To this end we derive the uncertainty relation for an arbitrary pair of observables and discuss the conditions for its validity. By means of a simple parameter-dependent state we illustrate the well known fact that the standard uncertainty relation for the coordinate and its conjugate momentum does not apply the pair of observables $ϕ-L_z$. This analysis is motivated by a discussion of the motion of an electron in a constant magnetic field appeared recently in this journal (Eur. J. Phys. \textbf{33} (2012) 1147) where the author assumed the validity of the standard uncertainty relation for the pair $ϕ-L_z$.

quant-ph↗
arXiv2012

Simple variational approaches to quantum wells

We discuss two simple variational approaches to quantum wells. The trial harmonic functions analyzed in an earlier paper give reasonable results for all well depths and are particularly suitable for deep wells. On the other hand, the exponential functions proposed here are preferable for shallow wells. We compare the shallow-well expansions for both kind of functions and show that they do not exhibit the cubic term appearing in the exact series. It is also shown that the deep-well expansion for the harmonic functions agree with the first terms of perturbation theory.

quant-ph↗
arXiv2011

Prony's method and the connected-moments expansion

We show that Prony's method provides the full solution to the nonlinear equations of the connected--moments expansion (CMX). Knowledge of all the parameters in the CMX ansatz is useful for the analysis of the convergence properties of the approach. Prony's method is also suitable for the calculation of the correlation function for simple quantum--mechanical models.

quant-ph↗
arXiv2011

Wronskian method and the Schrödinger eigenvalue march

We compare the Wronskian method (WM) and the Schrödinger eigenvalue march or canonical function method (SEM--CFM) for the calculation of the energies and eigenfunctions of the Schrödinger equation. The Wronskians between linearly independent solutions of the Schrödinger equation provide a rigorous basis for some of the assumptions of the SEM-CFM, like, for example, the concept of "saturation". We compare the performance of both approaches on a simple one-dimensional model and suggest that taking into account the asymptotic behaviour of the wavefunction (as is already done in the WM) may make the SEM-CFM more efficient from a numerical point of view.

quant-ph↗
arXiv2011

Solution to the Equations of the Moment Expansions

We develop a formula for matching a Taylor series about the origin and an asymptotic exponential expansion for large values of the coordinate. We test it on the expansion of the generating functions for the moments and connected moments of the Hamiltonian operator. In the former case the formula produces the energies and overlaps for the Rayleigh-Ritz method in the Krylov space. We choose the harmonic oscillator and a strongly anharmonic oscillator as illustrative examples for numerical test. Our results reveal some features of the connected-moments expansion that were overlooked in earlier studies and applications of the approach.

quant-ph↗
arXiv2011

Further analysis of the connected moments expansion

We apply the connected moments expansion to simple quantum--mechanical examples and show that under some conditions the main equations of the approach are no longer valid. In particular we consider two--level systems, the harmonic oscillator and the pure quartic oscillator.

quant-ph↗
arXiv2011

Resonances for symmetric two-barrier potentials

We describe a method for the accurate calculation of bound-state and resonance energies for one-dimensional potentials. We calculate the shape resonances for symmetric two-barrier potentials and compare them with those coming from the Siegert approximation, the complex scaling method and the box-stabilization method. A comparison of the Breit-Wigner profile and the transmission coefficient about its maximum illustrates that the agreement is better the sharper the resonance.

quant-ph↗