Local approximation to the critical parameters of quantum wells

Fernández, Francisco M.; García, Javier

doi:10.1016/j.amc.2013.06.049

Cited by 8 publications

(12 citation statements)

References 27 publications

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“…, λ n and allows to obtain it in terms of the others. For example, in the case of the one-dimensional Schrödinger equation, if the Hamiltonian depends on one parameter, then one can obtain the energy in terms of it, or the values of said parameter for which the energy adopts a particular value, as in the case of the critical parameters [45]. In the case of coupled equations (which is of concern in the present work), one should have as many coupled equations as unknown parameters, and Eq.…”

Section: The Methodsmentioning

confidence: 99%

Highly accurate potential energy curves for the hydrogen molecule ion

Fernández,

Garcia

2021

Preprint

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Potential energy surfaces of the hydrogen molecular ion H + 2 in the Born-Oppenheimer approximation are computed by means of the Riccati-Padé method (RPM). The convergence properties of the method are analyzed for different states. The equilibrium internuclear distance, as well as the corresponding electronic plus nuclear energy, and the associated separation constants, are computed to 40 digits of accuracy for several bound states. For the ground state the same parameters are computed with more than 100 digits of accuracy.Additional benchmark values of the electronic energy at different internuclear distances are given for several additional states. The software implementation of the RPM is given under a free software license. The results obtained in the present work are the most accurate available so far, and further additional benchmarks are made available through the software provided.

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Section: The Methodsmentioning

confidence: 99%

Highly accurate potential energy curves for the hydrogen molecule ion

Fernández,

Garcia

2021

Preprint

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“…It has been shown that the quantization condition ( 9) is consistent with moving a zero of ψ(x) towards infinity either along the real axis [13,14] or along a ray xe iβ on the complex coordinate plane [15]. In order to appreciate the latter statement clearer consider the canonical transformation…”

Section: The Riccati-padé Methodsmentioning

confidence: 99%

On two different kinds of resonances in one-dimensional quantum-mechanical models

Fernández

García

2016

J Math Chem

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We apply the Riccati-Padé method and the Rayleigh-Ritz method with complex rotation to the study of the resonances of a one-dimensional well with two barriers. The model exhibits two different kinds of resonances and we calculate them by means of both approaches. While the Rayleigh-Ritz method reveals each set at a particular interval of rotation angles the Riccati Padé method yields both of them as roots of the same Hankel

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“…The first systematic calculation of λ c for a variety of bound states of the H atom under SCP was performed by Rogers et al [43] utilizing a numerical differentiation method. The accuracy of these critical values was significantly improved by Diaz et al [44] using a matrix propagation technique and by Fernández and Garcia [45] based on the Riccati-Padé method. Our recent work [16] applied the 'indirect' generalized pseudospectral method (GPS) and extended such an effort by establishing what are so far the most accurate (near the precision of numeric arithmetic) prediction of critical screening parameters for the low-lying (n ⩽ 10) bound states of the H atom under SCP, HP, and ECSCP.…”

Section: Introductionmentioning

confidence: 99%

Critical screening parameters of one-electron systems with screened Coulomb potentials: circular Rydberg states

Jiao

Liu

et al. 2023

J. Phys. B: At. Mol. Opt. Phys.

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The critical screening parameters (λc) of one-electron systems in circular states under short-range screened Coulomb potentials are investigated using the generalized pseudospectral method. High-precision values of λc are obtained for bound states with the principal quantum number n up to a thousand. The polynomial ﬁttings based on the numerical values indicate that the high-Rydberg limits of the n^2-scaled critical screening parameters, i.e., n^2λc, are equal to 2/e, 1.295220475783830, and 0.543564037493402, respectively, for the Debye-Hückel, Hulthén, and exponential cosine screened Coulomb potentials. It is surprisingly found that these high-Rydberg limits can be well-reproduced from Bohr's correspondence principle together with the semi-classical model of the hydrogen atom. We further show that both the present numerical calculations and the semi-classical analysis can be extended to other short-range potentials such as the generalized exponential, Lennard-Jones, modiﬁed Pöschl‒Teller, shifted Morse, Woods-Saxon, and Manning‒Rosen potentials that attract wide interest in atomic, molecular, and nuclear physics.

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Local approximation to the critical parameters of quantum wells

Cited by 8 publications

References 27 publications

Highly accurate potential energy curves for the hydrogen molecule ion

Highly accurate potential energy curves for the hydrogen molecule ion

On two different kinds of resonances in one-dimensional quantum-mechanical models

Critical screening parameters of one-electron systems with screened Coulomb potentials: circular Rydberg states

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