A High-Precision Study of Anharmonic-Oscillator Spectra

Macfarlane, M.H.

doi:10.1006/aphy.1998.5854

Cited by 22 publications

(24 citation statements)

References 61 publications

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“….., given two initial values, Ψ 0 and Ψ 1 . Equation (13) leads to the following equation known as the energy quantification condition: Ψ n = 0. The energy levels are obtained by the energy values for which the curve of Ψ n (E) meets the energy axis.…”

Section: B Numerov Methodsmentioning

confidence: 99%

“…1 They have been carried out using the Hill determinant, 2,3 the coupled cluster method, 4,5 the Bargmann representation, 6,7 the variational-perturbation expansion and other approaches. [8][9][10][11][12][13] Using the double exponential Sinc collocation method, Gaudreau et al 14 evaluated the energy eigenvalues of anharmonic oscillators. For bounded anharmonic oscillators, Alhendi et al 15 used a power-series expansion and Fernández 16 applied the Riccati-Padé method, to calculate their accurate eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

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Airy function approach and Numerov method to study the anharmonic oscillator potentials V(x) = Ax2α + Bx2

Sdran

Maiz

2016

AIP Advances

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The numerical solutions of the time independent Schrödinger equation of different one-dimensional potentials forms are sometime achieved by the asymptotic iteration method. Its importance appears, for example, on its efficiency to describe vibrational system in quantum mechanics. In this paper, the Airy function approach and the Numerov method have been used and presented to study the oscillator anharmonic potential V(x) = Ax2α + Bx2, (A>0, B<0), with (α = 2) for quadratic, (α =3) for sextic and (α =4) for octic anharmonic oscillators. The Airy function approach is based on the replacement of the real potential V(x) by a piecewise-linear potential v(x), while, the Numerov method is based on the discretization of the wave function on the x-axis. The first energies levels have been calculated and the wave functions for the sextic system have been evaluated. These specific values are unlimited by the magnitude of A, B and α. It’s found that the obtained results are in good agreement with the previous results obtained by the asymptotic iteration method for α =3.

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Section: B Numerov Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Airy function approach and Numerov method to study the anharmonic oscillator potentials V(x) = Ax2α + Bx2

Sdran

Maiz

2016

AIP Advances

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“…Anharmonicity plays a key role in studies of molecular vibrations, quantum oscillations and semiconductor engineering. High precision not only handled the highly singular linear systems involved, but also prevented convergence to the wrong eigenvalue [33].…”

Section: Anharmonic Oscillatorsmentioning

confidence: 99%

High-Precision Arithmetic in Mathematical Physics

Bailey

Borwein

2015

Mathematics

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For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This article discusses the challenge of high-precision computation, in the context of mathematical physics, and highlights what facilities are required to support future computation, in light of emerging developments in computer architecture.

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“…Via a logarithmic-derivative-like transform (8), one is face to the problem of solving a relatively simple nonlinear ODE the analytic properties of which are well controlled by those of ψ (x) solution of the linear second order ODE (1). This gives the opportunity of studying how the various quasi-analytical methods for solving nonlinear ODE (presented below) actually work.…”

Section: The Nonlinear Odementioning

confidence: 99%

Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

Bervillier¹

2009

J. Phys. A: Math. Theor.

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Abstract. The simplicity and the efficiency of a quasi-analytical method for solving nonlinear ordinary differential equations (ODE), is illustrated on the study of anharmonic oscillators (AO) with a potential V (x) = βx 2 + x 2m (m > 0). The method [Nucl. Phys. B801, 296 (2008)], applies a priori to any ODE with two-point boundaries (one being located at infinity), the solution of which has singularities in the complex plane of the independent variable x. A conformal mapping of a suitably chosen angular sector of the complex plane of x upon the unit disc centered at the origin makes convergent the transformed Taylor series of the generic solution so that the boundary condition at infinity can be easily imposed. In principle, this constraint, when applied on the logarithmic-derivative of the wave function, determines the eigenvalues to an arbitrary level of accuracy. In practice, for β ≥ 0 or slightly negative, the accuracy of the results obtained is astonishingly large with regards to the modest computing power used. It is explained why the efficiency of the method decreases as β is more and more negative. Various aspects of the method and comparisons with some seemingly similar methods, based also on expressing the solution as a Taylor series, are shortly reviewed, presented and discussed.

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A High-Precision Study of Anharmonic-Oscillator Spectra

Cited by 22 publications

References 61 publications

Airy function approach and Numerov method to study the anharmonic oscillator potentials V(x) = Ax2α + Bx2

Airy function approach and Numerov method to study the anharmonic oscillator potentials V(x) = Ax2α + Bx2

High-Precision Arithmetic in Mathematical Physics

Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

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