Asymptotic Functional Form Preservation Method for the Perturbation Theory: Application to Anharmonic Oscillators

2016 ◽  
Vol 63 (10) ◽  
pp. 873-885 ◽  
Author(s):  
Chia-Chun Chou ◽  
Ching-Teh Li
Keyword(s):  
Perturbation Theory ◽  
Functional Form ◽  
Anharmonic Oscillators ◽  
Preservation Method ◽  
Theory Application

Asymptotic Functional Form Preservation Method with Supersymmetric Quantum Mechanics for Anharmonic Oscillators

2016 ◽  
Vol 69 (9) ◽  
pp. 950 ◽  
Author(s):  
Chia-Chun Chou ◽  
Ching-Teh Li
Keyword(s):  
Quantum Mechanics ◽  
Excited State ◽  
Series Expansion ◽  
Functional Form ◽  
Full Range ◽  
Supersymmetric Quantum Mechanics ◽  
Anharmonic Oscillators ◽  
Approximate Analytical Expression ◽  
Preservation Method ◽  
Functional Forms

The asymptotic functional form preservation method is developed in the framework of supersymmetric quantum mechanics to obtain the energies and wave functions of anharmonic oscillators. For each of the ground states in the hierarchy of supersymmetric partner Hamiltonians, we derive a series expansion of the superpotential for . Employing a transformation containing an unphysical parameter, we convert the series expansion of the superpotential into a new series expansion applicable to all the range of x. The unphysical parameter is determined by the principle of minimal sensitivity. Requiring the preservation of the correct asymptotic functional form of the full-range series expansion as x tends to infinity, we obtain the ground and excited state energies. The truncated full-range series expansion for the superpotential provides an approximate analytical expression for the wave function. In addition, several ansatz functional forms are also proposed for the superpotential to obtain the ground and excited state energies with high accuracy. Excellent computational results for the anharmonic oscillator demonstrate that the method proposed here is suitable for solving similar quantum mechanical problems.

scholarly journals Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

2020 ◽  
Vol 35 (01) ◽  
pp. 2050005
Author(s):  
J. C. del Valle ◽  
A. V. Turbiner
Keyword(s):  
Perturbation Theory ◽  
Anharmonic Oscillator ◽  
Strong Coupling Regime ◽  
Weak Coupling Regime ◽  
Fractional Powers ◽  
Anharmonic Oscillators ◽  
Relative Deviation ◽  
Variational Calculations ◽  
Compact Approximation ◽  
Semiclassical Expansion

In our previous paper I (del Valle–Turbiner, 2019) a formalism was developed to study the general [Formula: see text]-dimensional radial anharmonic oscillator with potential [Formula: see text]. It was based on the Perturbation Theory (PT) in powers of [Formula: see text] (weak coupling regime) and in inverse, fractional powers of [Formula: see text] (strong coupling regime) in both [Formula: see text]-space and in [Formula: see text]-space, respectively. As a result, the Approximant was introduced — a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variational calculations, it has led to variational energies of unprecedented accuracy for cubic anharmonic oscillator. In this paper, the formalism is applied to both quartic and sextic, spherically-symmetric radial anharmonic oscillators with two term potentials [Formula: see text], [Formula: see text], respectively. It is shown that a two-parametric Approximant for quartic oscillator and a five-parametric one for sextic oscillator for the first four eigenstates used to calculate the variational energy are accurate in 8–12 figures for any [Formula: see text] and [Formula: see text], while the relative deviation of the Approximant from the exact eigenfunction is less than [Formula: see text] for any [Formula: see text].

On nonintegral E corrections in perturbation theory: application to the perturbed Morse oscillator

1994 ◽  
Vol 72 (1-2) ◽  
pp. 80-85 ◽  
Author(s):  
Hafez Kobeissi ◽  
Majida Kobeissi ◽  
Chafia H. Trad
Keyword(s):  
Perturbation Theory ◽  
Good Accuracy ◽  
Morse Oscillator ◽  
Numerical Application ◽  
Correction Formula ◽  
Theory Application ◽  
Nonhomogeneous Differential Equation ◽  
Particular Solution ◽  
New Formulation ◽  
Schrödinger Perturbation

A new formulation of the Rayleigh–Schrödinger perturbation theory is applied to the derivation of the vibrational eigenvalues of the perturbed Morse oscillator (PMO). This formulation avoids the conventional projection of the Ψ corrections on the basis of unperturbed eigenfunctions [Formula: see text], or the projection of the nonhomogeneous Schrödinger equations on [Formula: see text], it gives simple expressions for each E correction [Formula: see text] free of summations and integrals. When the PMO is characterized by the potential U = UM + UP (where UM is the unperturbed Morse potential), the eigenvalue of a vibrational level ν is given by: [Formula: see text]. According to the new formulation the correction £, [Formula: see text] is given by [Formula: see text], where σp(r) is a particular solution of the nonhomogeneous differential equation y″ + f y = sp; here [Formula: see text], sp is well known for each p: for p = 0, [Formula: see text]; for [Formula: see text]. For the numerical application one single routine is used, that of integrating y″ + f y = s, where the coefficients are known as well as the initial values. An example is presented for the Huffaker PMO of the (carbon monoxide) CO-X1Σ+ state. The vibrational eigenvalues Eν are obtained to a good accuracy (with p = 4) even for high levels. This result confirms the validity of this new formulation and gives a semianalytic expression for the PMO eigenvalues.

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