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IOP Conference Series Earth and Environmental Science

Analytic ground state energy and wave function of a quantum system in an exponential-form anharmonic interaction potential

Israel B Babalola, Oluwole Emmanuel Oyewande, Idowu H Latok, Olukunle Charles Olawole

Published Feb 1, 2021• 0 citations
Publisher

Abstract

Abstract An anharmonic oscillator with a perturbed quadratic potential which is coupled with an exponential term is being investigated in this paper by isolating an anharmonic oscillator interaction potential from the actual interaction experienced by the quantum system studied, and using standard perturbative techniques. The anharmonic potential considered is of interest because of its usefulness in the study of non-centrosymmetric materials which have applications in piezoelectricity. The ground state energy eigenvalue and its associated eigenstates were calculated for the quantum system using an analytical approach. Results obtained are compared to those of quantised harmonic oscillator to show the effect of the perturbation.

DOI
10.1088/1755-1315/655/1/012047

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Abstract An anharmonic oscillator with a perturbed quadratic potential which is coupled with an exponential term is being investigated in this paper by isolating an anharmonic oscillator interaction potential from the actual interaction experienced by the quantum system studied, and using standard perturbative techniques. The anharmonic potential considered is of interest because of its usefulness in the study of non-centrosymmetric materials which have applications in piezoelectricity. The ground state energy eigenvalue and its associated eigenstates were calculated for the quantum system using an analytical approach. Results obtained are compared to those of quantised harmonic oscillator to show the effect of the perturbation.
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This work is aimed at obtaining the energy eigenvalues for one-dimensional quantum harmonic and anharmonic oscillators perturbed by linear, quadratic, cubic and polynomial potentials. To obtain the solutions of the energy eigenvalues, we employed the time-independent perturbation theory to calculate the first and the second-order energy correction, which we used to obtain the complete generalised energy eigenvalues of the quantum harmonic oscillators with linear, quadratic, cubic and polynomial perturbation potential of the same unperturbed Hamiltonian (H).
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Abstract The quantum anharmonic oscillator has been solved numerically using matrix diagonalization technique. The interaction potential consisting of quadratic ( 1 2 k x 2 ) and quartic ( λx 4 ) terms is embedded within an infinite square well potential of appropriate width, ‘ a ’ and its sine eigen functions are used as basis functions ‘ N ’ for the employed matrix method. The energy eigen values for the resultant Hamiltonian are solved in a free open source software (FOSS), Gnumeric, a simple worksheet environment. The numerical parameters ‘ a ’ and ‘ N ’ are optimized to converge to the expected energies for harmonic oscillator and those for anharmonic oscillator from perturbation theory for small values of physical parameter, ‘ λ ’. The pure quartic oscillator is studied for both small and large values of λ and validated with results obtained from other numerical techniques. The breakdown of perturbation approximation for large values of λ is also shown.
In a recent paper [L. Ferrari, Phys. Rev. A 57, 2347 (1998)], one of the authors has shown that the mean energy of a quantum oscillator with periodic time-dependent frequency diverges exponentially in time, under certain conditions. In the present paper, we study the explicit form of the evolving state, and compare the results obtained with the general expressions developed by other authors for an arbitrary time-dependent frequency. Then we approach the problem of the anharmonic effects. A first-order calculation is performed in the case of a short-range perturbation. The transition rates between different states, evolving with the unperturbed Hamiltonian, are shown to vanish at long times when the unperturbed oscillator's energy diverges exponentially. A nonperturbative approach must be adopted, in the presence of anharmonic potentials of the form $V(q)\ensuremath{\propto}{q}^{j}, j>2$. In the case of weak anharmonicity $(j\ensuremath{-}2\ensuremath{\ll}1)$, a mean-field procedure can be used to show that the mean energy does actually saturate at long times, with the possible exception of periodic peaks, having nonsaturating height, that we call ``special quantum effects.''
In this work, an unusual approach is taken to calculate the ground state and first excited state energies of the anharmonic oscillator potential. By using the variational method, analytical wave functions which have extremely accurate expectation values for the quartic or sextic oscillators are obtained. The corrections due to the other terms are then calculated perturbatively. It is shown that each approach, which starts with a different choice of zeroth‐order term, provides results that agree with those computed from numerical integration and series solution. Joining the above schemes, yields a combined variational and perturbative treatment of anharmonic oscillator problems.