Power-Series Solutions for Energy Eigenvalues

Secrest, Don; Cashion, K.; Hirschfelder, Joseph O.

doi:10.1063/1.1733169

Cited by 44 publications

(27 citation statements)

References 5 publications

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“…The Padé method [10] is also an easy-to-use method which is more robust but less refined than the mapping method. The two methods (and also the analytic-continuation method [15,16]) may certainly be advantageously associated in the process of solving a two-point boundary problem of a nonlinear ODE. With M = 115, R = 6.03344983950017 and α = 2 the first odd state has been determined with 70 significant figures:…”

Section: Discussionmentioning

confidence: 99%

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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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Section: Discussionmentioning

confidence: 99%

“…In that case Secrest et al [15] have suggested to reach the value z 0 in more than one step using the Taylor expansion about a non-zero value z l < z 0 . This suggestion is already the analyticcontinuation method introduced later on by Holubec and Stauffer [16].…”

Section: The Power-series Methodsmentioning

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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“…p͑x͒ ϭ 1 q͑x͒ ϭ Ϫ7x 2 ϩ 0.5x 3 ϩ x 4 w͑x͒ ϭ 0.5 a ϭ Ϫϱ LPN (36) * Morse potential [Secrest et al 1962]. (37) Quartic anharmonic oscillator [Scott et al 1969].…”

Section: Infinite Interval Singular: Lpn and Lcn-friedrichs Bcmentioning

confidence: 99%

A test package for Sturm-Liouville solvers

Pryce¹

1999

ACM Trans. Math. Softw.

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The author and colleagues have produced a collection of 60 test problems which offer a realistic performance test of the currently available automatic codes for eigenvalues of the classical Sturm-Liouville problem. We describe a Fortran implementation and the considerations that went into its design. A novel feature is that (almost) all the code defining one problem is textually contiguous in the Fortran text, unlike for example the DETEST package for ODE initial-value solvers where the definition of a problem is spread over several routines. The described implementation forms the infrastructure of the SLDRIVER interactive package which supports exploration of a set of Sturm-Liouville problems with the four SL-solvers SLEIGN, SLEDGE, SL02F, and SLEIGN2. A "standard" set of 60 problems is provided, but it is simple to replace this by another one.

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“…The double-well potential problem and its generalizations have long been the object of several theoretical studies [1][2][3][4][5][6][7][8] in view of their importance as models for exchange forces, 2,3 and possible applications to real physical systems such as H 2 ϩ and NH 3 as well as models for H-bonding.…”

Section: Introductionmentioning

confidence: 99%

Exchange energy in a double-well potential profile from fluctuation theory

Battezzati

Magnasco

2001

The Journal of Chemical Physics

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Articles you may be interested inOn a solution of the Schrödinger equation with a hyperbolic double-well potential Helium mediated deposition: Modeling the He−TiO2(110)-(1×1) interaction potential and application to the collision of a helium droplet from density functional calculations Quantum Monte Carlo calculations of the potential energy curve of the helium dimer This article shows that the asymptotic quantum mechanical exchange energies and wave functions in a bistable potential can be obtained by computing the complex-valued first-passage time across the potential barrier from a transient state endowed with Smoluchowski boundary conditions, whose probability density is concentrated on one side of the barrier. This interpretation is validated by one of the authors' previous work on diffusion in a random field, showing that this model yields a diffusion equation equivalent to quantum mechanical equations.

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Power-Series Solutions for Energy Eigenvalues

Cited by 44 publications

References 5 publications

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

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

A test package for Sturm-Liouville solvers

Exchange energy in a double-well potential profile from fluctuation theory

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