Development of the perturbation theory using polynomial solutions

Abstract

The number of quantum systems for which the stationary Schrodinger equation is exactly solvable is very limited. These systems constitute the basic elements of the quantum theory of perturbation. The exact polynomial solutions for real quantum potential systems provided by the use of Lagrange interpolation allows further development of the quantum perturbation theory. In fact, the first order of correction for the value of the energy appears to be sufficient since the chosen perturbation Hamiltonian is very small or even negligible compared to the main Hamiltonian. Here, we use the perturbation theory to derive polynomial solutions, and we then find that our approximated results agree very well with previous published or numerically achieved ones. We believe that this study is an operational tool for the verification and improvement of numerical and approximate methods.

Development and refinement of the Variational Method based on Polynomial Solutions of Schrödinger Equation

“…The obtained results for the ground state energy values are found to be accurate. Finally, our results agreed well with those available in literature [27,30,31]. We believe that this study will encourage the researchers and scientists for further investigation of general polynomial potentials.…”

“…)︁ . These results were found to be the same as previously published for the same potential energy and level [27].…”

“…By limiting the calculations in the vicinity of the previous coefficients a i=1,2,3,4 , we were successful in reducing computation time. Finally, we deduce the estimated ground state energy value: ,2,3,4 ) and compare this result with available ones for the same potential energy [27].…”