A simple accurate formula for the energy levels of oscillators with a quartic potential

Mathews, P. M.; Seetharaman, M.; Raghavan, S.; Bhargava, Valmik

doi:10.1016/0375-9601(81)90511-9

Cited by 23 publications

(4 citation statements)

References 2 publications

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“…We will show that both energies and wave functions will be represented by closed analytic expressions with the accuracy of the wave functions being be-tween 0.1 and 1 percent for both small and large coupling constants. Various accurate analytic expressions for the energies have already appeared in the literature based on using convergent, strong coupling expansions generated by rearrangement of the usual divergent weak coupling expansion [28] or by some variational requirement [29]. However, accurate analytic expressions representing wave functions have not hitherto been known.…”

Section: Introductionmentioning

confidence: 99%

Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators

Liverts

Mandelzweig

Tabakin

2006

Journal of Mathematical Physics

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Ground state energies and wave functions of quartic and pure quartic oscillators are calculated by first casting the Schrödinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is solved by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. Our explicit analytic results are then compared with exact numerical and also with WKB solutions and it is found that our ground state wave functions, using a range of small to large coupling constants, yield a precision of between 0.1 and 1 percent and are more accurate than WKB solutions by two to three orders of magnitude. In addition, our QLM wave functions are devoid of unphysical turning point singularities and thus allow one to make analytical estimates of how variation of the oscillator parameters affects physical systems that can be described by the quartic and pure quartic oscillators.

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Section: Introductionmentioning

confidence: 99%

Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators

Liverts

Mandelzweig

Tabakin

2006

Journal of Mathematical Physics

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“…Since perturbation theory breaks down, a variety of non-perturbative approximation methods have been employed to study them. Some of the methods are WKB approximation [1,2], Hill determinant method [3], action-angle technique [4], Continued fraction method [5], the variational method [6,7], Re-normalised frequency method [8] Chebyshev polynomial method [9], the residue-squaring method [10], Pade approximants method [11], the kinetic potential method [12], the fixed point method [13], the hypervirial method [14] and so on. One of the simplest methods was proposed by Ginsberg and Montroll [15].…”

Section: Introductionmentioning

confidence: 99%

Ground State Energy Eigenvalues of an Octic Oscillator with Quartic and Sextic Anharmonicities

2016

IJSR

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“…Oscillators are an essential component in devices in electron positron collider systems (see, e.g., Zhao et al [1], Ma et al [2], Zang et al [3], Ding et al [4], Marder et al [5], Barroso [6], Miller et al [7], and Lemke [8], just citing a few). As a matter of fact, oscillations are phenomena widely observed in sciences and engineering relating to high energy physics (see, e.g., Akhmediev et al [9], Bachas [10], Winter et al [11], Dodonov [12], Tan [13], Diamandis et al [14], Greenwald et al [15], Mathews et al [16], Faiman [17], Cocho et al [18], Baldiotti et al [19], Kyu Shin [20], Kirson [21], Clement [22], Sikström et al [23], Asghari et al [24], Um et al [25], Bahar and Yasuk [26], Hassanabadi et al [27], Bhattacharya and Roy [28], and Saad et al [29], simply mentioning a few).…”

Section: Introductionmentioning

confidence: 99%

Characteristic Roots of a Class of Fractional Oscillators

Lim

Cattani

et al. 2013

Advances in High Energy Physics

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The fundamental theorem of algebra determines the number of characteristic roots of an ordinary differential equation of integer order. This may cease to be true for a differential equation of fractional order. The results given in this paper suggest that the number of the characteristic roots of a class of oscillators of fractional order may in general be infinitely great. Further, we infer that it may also be the case for the characteristic roots of a differential equation of fractional order greater than 1. The relationship between the range of the fractional order and the locations of characteristic roots of oscillators in the complex plane is considered.

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A simple accurate formula for the energy levels of oscillators with a quartic potential

Cited by 23 publications

References 2 publications

Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators

Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators

Ground State Energy Eigenvalues of an Octic Oscillator with Quartic and Sextic Anharmonicities

Characteristic Roots of a Class of Fractional Oscillators

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