Spectrum of one-dimensional anharmonic oscillators

We show that for the triple well potential with non-equivalent vacua, instantons generate for the low lying energy states a singlet and a doublet of states rather than a triplet of equal energy spacing. Our energy splitting formulas are also confirmed numerically. This splitting property is due to the presence of non-equivalent vacua. A comment on its generality to multi-well is presented.Instantons are non-trivial classical solutions of Euclidean field equations for which the action is finite. 1 Their importance, besides being topological configurations, comes from their finite contributions to Feynman path integral. In quantum mechanics instantons correspond to non-trivial finite classical solutions of classical equations of motion with inverted potential. 2 The uses of instanton calculations have proven to be useful in analyzing nonperturbative aspects of quantum mechanical systems with degenerate vacua, this is because instanton solutions contribute to the quantum tunnelling phenomena, which can be calculated with the aid of the dilute gas approximation. A celebrated example is the splitting of energy level in the symmetric double well case. 2 To our knowledge only a few recent papers attempted to apply the instanton method to the triple well potential with non-equivalent vacua. 3-5 This problem is rather involved compared to the symmetric double well case. It has been suspected that in the presence of non-equivalent vacua the dilute gas approximation may break down. 5 Moreover, it has been claimed 3,4 that the average of the harmonic frequencies over the non-equivalent vacua of the potential serves as the central § Corresponding author 2103 Mod. Phys. Lett. A 2004Lett. A .19:2103Lett. A -2112. Downloaded from www.worldscientific.com by UNIVERSITY OF BRITISH COLUMBIA on 10/24/12. For personal use only.

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“…Moreover, our results are consistent with what one could predict based on simple quantum mechanical approach to the problem Ref. 10.…”

Section: Symmetric Triple Well With Non-equivalent Vacua 2107supporting

confidence: 92%

“…(10), and taking into account the correct counting together with the starting initial integral I(0, 0) = B δ [e δ T 2 − e −δ T 2 ], we get (here we keep the common factor in the definition of I(n, m))…”

Section: Symmetric Triple Well With Non-equivalent Vacua 2107mentioning

confidence: 99%

Symmetric Triple Well With Non-Equivalent Vacua: Instantonic Approach

Alhendi

Lashin

2004

Mod. Phys. Lett. A

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“…This is, of course, the usual way that series solutions of second-order linear homogeneous ODE are normally discovered. As a first example, we consider the following Schrödinger equation (Alhendi and Lashin [1]…”

Section: W Robinmentioning

confidence: 99%

“…In section 2 we solve (1.1) for the case of 0 z being an ordinary point and present the general series solution for ) (z P and ) (z Q arbitrary analytic functions; the algorithm is then exemplified using two problems requiring the solution of the Schrödinger equation [1,11]. In section 3 we solve (1.1) for the case of 0 z being a regular singular point and present the general Frobenius series solution; the algorithm is then exemplified, again, via a problem requiring the solution of the Schrödinger equation [5].…”

Section: Introductionmentioning

confidence: 99%

On the procedure for the series solution of second-order homogeneous linear differential equation via the complex integration method

Robin¹

2014

nade

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The theory of series solutions for second-order linear homogeneous ordinary differential equation is developed ab initio, using an elementary complex integral expression (based on Herrera' work [3]) derived and applied in previous papers [8,9]. As well as reproducing the usual expression for the recurrence relations for second-order equations, the general solution method is straight-forward to apply as an algorithm on its own, with the integral algorithm replacing the manipulation of power series by reducing the task of finding a series solution for second-order equations to the solution, instead, of a system of uncoupled simple equations in a single unknown. The integral algorithm also simplifies the construction of 'logarithmic solutions' to second-order Fuchs, equations. Examples, from the general science and mathematics literature, are presented throughout.Mathematics Subject Classification: 30B10, 30E20 34A25, 34A30

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“…[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. 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.…”

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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Spectrum of one-dimensional anharmonic oscillators

Cited by 12 publications

References 15 publications

Symmetric Triple Well With Non-Equivalent Vacua: Instantonic Approach

Symmetric Triple Well With Non-Equivalent Vacua: Instantonic Approach

On the procedure for the series solution of second-order homogeneous linear differential equation via the complex integration method

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

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