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Analytical Solution Of Schrödinger Equation With Mie-Type Potential Using Factorisation Method

B. Okon, Eno E. Ituen, Oyebola O. Popoola, D Akaninyene Antia

Published Jan 1, 2013• 6 citations
OpenAlex

Abstract

we have obtained the analytical solution of Schrodinger wave equation with Mie – type potential using factorization method. We have also obtained energy eigenvalues of our potential and the corresponding wave function using an ansatz and then compare the result to standard Laguerre’s differential equation. Under special cases our potential model reduces two well known potentials such as Coulomb and the Kratzer Feus potentials.

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•Jan 1, 2012•0 citations
we have obtained the analytical solution of Schrodinger wave equation with Mie – type potential using factorization method. We have also obtained energy eigenvalues of our potential and the corresponding wave function using an ansatz and then compare the result to standard Laguerre’s differential equation. Under special cases our potential model reduces two well known potentials such as Coulomb and the Kratzer Feus potentials.
Exactly solvable models play an extremely important role in many fields of quantum physics. In this study, the Schrödinger equation is applied for a solution of a two--dimensional (2D) problem for two particles interacting via Kratzer, and modified Kratzer potentials. We found the exact bound state solutions of the two--dimensional Schrödinger equation with Kratzer--type potentials and present analytical expressions for the eigenvalues and eigenfunctions. The eigenfunctions are given in terms of the associated Laguerre polynomials.
The exact solutions of the N-dimensional Schrodinger equation with the Mie-type potentials are obtained using the conventional Nikiforov-Uvarov method.The expectation values r^{-1} and r^{-2}$ and the virial theorem are also obtained in N-dimensions using the Hellmann-Feynman theorem.The ladder operators are also construct for the Mie-type potentials in N-dimensions and the matrix elements of some operators rrr and r\frac{d}{dr} are analytically obtained from the ladder operators.And the general results reduce to the 3-dimensional case when N=3.
The general solutions of the Schrödinger equation for a non-central potential are obtained by using the Nikiforov–Uvarov method. The Schrödinger equation with general non-central potential is separated into radial and angular parts, and energy eigenvalues and eigenfunctions are derived analytically. By making special selections, the non-central potential is reduced to Coulomb and Hartmann ring-shaped potentials, and the obtained results are compared with the solutions of Coulomb and Hartmann potentials given in the literature.
The goal of the dissertation is to find new method of solving two-dimensional Schrödinger equation in such cases, when the separation of the variables is not applicable. The results are applied to the two-dimensional Schrödinger equation with the potentials of the form of the quartic polynomial, of the sextic polynomial and of the quartic Morse potential. For these cases, the analytical formulae for the ground state wave functions and the corresponding energies have been found. For the specific class of the potentials of the form of the quartic polynomial, analytical formula for one of the excited states and for the corresponding energy have been found.