Exact Solutions of the Morse-like Potential, Step-Up and Step-Down Operators via Laplace Transform Approach

TL;DR

This paper derives exact solutions for a Morse-like potential using Laplace transforms, revealing ladder operators that satisfy SU(2) algebra, and compares these results to the Morse potential.

Contribution

It introduces a method to find eigenfunctions and eigenvalues for a Morse-like potential and constructs ladder operators via Laplace transforms, linking them to SU(2) algebra.

Findings

Ladder operators satisfy SU(2) commutation relations.

Eigenfunctions and eigenvalues are obtained analytically.

Results are analogous to those of the Morse potential.

Abstract

We intend to realize the step-up and step-down operators of the potential $V(x)=V_{1}e^{2\beta x}+V_{2}e^{\beta x}$. It is found that these operators satisfy the commutation relations for the SU(2) group. We find the eigenfunctions and the eigenvalues of the potential by using the Laplace transform approach to study the Lie algebra satisfied the ladder operators of the potential under consideration. Our results are similar to the ones obtained for the Morse potential ($\beta \rightarrow -\beta$).