Exact Solutions of the Schr\"odinger Equation via Laplace Transform Approach: Pseudoharmonic potential and Mie-type potentials

TL;DR

This paper derives exact bound state solutions and eigenfunctions for the Schrödinger equation with pseudoharmonic and Mie-type potentials using Laplace transforms, confirming previous results and providing new numerical data for molecular potentials.

Contribution

It introduces an analytical Laplace transform method to solve the Schrödinger equation for specific potentials, extending existing solutions and including numerical results for molecular applications.

Findings

Exact solutions match previous results

Eigenfunctions and eigenvalues are explicitly derived

Numerical results for diatomic molecules are provided

Abstract

Exact bound state solutions and corresponding normalized eigenfunctions of the radial Schr\"odinger equation are studied for the pseudoharmonic and Mie-type potentials by using the Laplace transform approach. The analytical results are obtained and seen that they are the same with the ones obtained before. The energy eigenvalues of the inverse square plus square potential and three-dimensional harmonic oscillator are given as special cases. It is shown the variation of the first six normalized wavefunctions of the above potentials. It is also given numerical results for the bound states of two diatomic molecular potentials, and compared the results with the ones obtained in literature.