On the Nature of Energy Levels of Anharmonic Oscillators

Chebotarev, L. V.

doi:10.1006/aphy.1998.5902

Cited by 13 publications

(14 citation statements)

References 40 publications

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“…Differentiating (21) with respect to n via the Liebniz rule [25] and using (20), one obtains N neatly as N = 2m h dE dn (22) and notes, in passing, also its connection with the time period defined in (5): τ = 2m/N. The significance of N is thus quite apparent.…”

Section: Some General Results and Theoremsmentioning

confidence: 99%

“…Indeed, the agreement of WSQR II with exactness for the special case of N = 2 is usually highlighted in its favor, but the observed opposite trend is never pointed out. Very recently [20], however, this particular observation has been given the due importance to explore further modifications of WSQR II with respect to quantization of the action integral J, introducing ideas of complex phase-space trajectories and consequent complex turning points.…”

Section: Variants Of Wsqrmentioning

confidence: 99%

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Wilson-Sommerfeld quantization rule revisited

Mukhopadhyay

Bhattacharyya

Pathak

2001

Int. J. Quantum Chem.

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A fresh look at the origin of the Wilson–Sommerfeld quantization rule has been pursued to gain new insight. The rule is shown to provide states that satisfy several well‐known theorems of standard quantum mechanics. A few other useful results and scaling relations are also derived. They emerge to act as nice guiding rules of thumb in the course of rigorous computations. Certain features of true excited‐state densities can be understood. Goodness of approximate densities can be assessed. Compressed systems can be studied profitably. A route is also sketched that allows one to retrieve classical trajectories from near‐exact energy eigenfunctions for both bound and resonant states by exploiting this rule. Additionally, a discussion on semiclassical perturbation theory is presented emphasizing the asymptotic behavior. Pilot calculations demonstrate the success of the present endeavor under various circumstances. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 113–125, 2001

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Section: Some General Results and Theoremsmentioning

confidence: 99%

Section: Variants Of Wsqrmentioning

confidence: 99%

Wilson-Sommerfeld quantization rule revisited

Mukhopadhyay

Bhattacharyya

Pathak

2001

Int. J. Quantum Chem.

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“…A large number of workers have studied various symmetrical 1‐D potentials that can be expressed as the ratio of two polynomials in x 2 . In particular, the GAO problem 26–31 has been studied in great detail using an assortment of techniques, including the MEP 9–11. We now briefly discuss how such potentials can be treated in our present scheme.…”

Section: Formalismmentioning

confidence: 99%

“…The layout of the article is as follows: In Section 2 we present the basic methodology, and then outline a route for studying excited states in Section 3. The application of our strategy to (1) the hydrogen atom in strong radial electric 18–20, magnetic 5, 21–24, and combined fields 25 and (2) the general anharmonic oscillator (GAO) problem 26–31 will be discussed in Section 4. We summarize our findings in Section 5.…”

Section: Introductionmentioning

confidence: 99%

Exploiting the interdependence of maximum entropy parameters in eigenvalue problems

Bandyopadhyay

Bhattacharyya

Bhattacharya

2002

Int J of Quantum Chemistry

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ABSTRACT:We exploit the interrelation among the parameters embedded in the maximum entropy ansatz to develop a scheme for obtaining accurate estimates of the ground-state energy and wave function of systems for which the potential is represented by a rational function. Our scheme reduces an N-parameter optimization problem to a two-parameter one, leading to considerable simplification of the prevalent strategy. An indirect route for the study of excited states is also sketched. Test calculations on hydrogenic systems subject to strong or superstrong radial magnetic fields with and without electric field reveal the advantages of our approach. Additional studies on 1-D anharmonic oscillators affirm its workability and generality.

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“…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. A variant of approximate methods and numerical techniques has recently been developed, [17][18][19][20][21][22][23][24] to calculate with high precision, the spectrum of one-dimensional symmetric anharmonic oscillators. Barakat 25 used the asymptotic iteration method (AIM) to calculate the eigen-energies for the anharmonic oscillator potentials V(x) = Ax 2α + Bx 2 , he introduced an adjustable parameter β to improve the AIM rate of convergence.…”

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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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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On the Nature of Energy Levels of Anharmonic Oscillators

Cited by 13 publications

References 40 publications

Wilson-Sommerfeld quantization rule revisited

Wilson-Sommerfeld quantization rule revisited

Exploiting the interdependence of maximum entropy parameters in eigenvalue problems

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

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