Abstract
Abstract The variational method is known as a powerful and preferred technique to find both analytical and numerical solutions for numerous forms of anharmonic oscillator potentials. In the present study, we considered certain conditions for the choice of the trial wave function. The current form of the trial wave function is based on the possible polynomial solutions of the Schrödinger equation. The advantage of our modified variational method is its ability to reduce the calculation steps and hence computation time. Also, we compared the results provided by our modified method with the results obtained by different methods in general but particularly Numerov method for the same problem.
Highlights
The precise solution of the Schrodinger equation is possible only in few cases such as infinite square well and harmonic oscillator potentials
The current form of the trial wave function is based on the possible polynomial solutions of the Schrödinger equation
We present some insight from available literature about variational principle together with appropriate approximations for the electron-electron interactions which are the basis for most practical approaches to solving the Schrödinger equation in condensed matter physics
Summary
The precise solution of the Schrodinger equation is possible only in few cases such as infinite square well and harmonic oscillator potentials. Popescu et al [6] considered a different form of the successive variational method based on a solution of a differential equation They successfully combined the variational method which uses variational global parameter with the finite element method for the study of the generalized anharmonic oscillator in D dimensions [7]. Cooper et al [8] in another work used a newly suggested algorithm of Gozzi, Reuter and, Thacker to determine the excited states of one-dimensional systems They determined approximated eigenvalues and eigenfunctions of the anharmonic oscillator. F M Fernández and J Garcia [20] considered Rayleigh-Ritz variational computations with non-orthogonal basic sets with the correct asymptotic behavior This approach is illustrated by the construction of appropriate basis sets for one-dimensional models such as the two double-well oscillators recently examined by other authors.
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