# Role of conditional shape invariance symmetry property to obtain eigen-spectrum of the generalized polynomial potential with a Coulomb term

## TL;DR

This paper introduces a supersymmetric quantum mechanics method leveraging conditional shape invariance to analytically and numerically determine eigenvalues of generalized polynomial potentials with Coulomb terms, including excited states.

## Contribution

It develops a new analytical approach using conditional shape invariance in SUSY quantum mechanics for solving eigen-spectrum problems with Coulomb-modified polynomial potentials.

## Key findings

- Analytical solutions for ground and excited states using the new method.
- Numerical eigenvalues closely match analytical results.
- Method applicable to quartic and sextic polynomial potentials.

## Abstract

A method based on supersymmteric (SUSY) quantum mechanics has been developed by exploiting conditional Shape invariance property for obtaining exact ground state solution of generalized polynomial potential with Coulomb term. Specific cases have been discussed with extensive analytical calculation. How this method can be used to calculate the excited states has also been demonstrated. We have also used a numerical technique (RKGS) and obtained the energy eigenvalues upto second excited state by solving the Schrodinger equation for quartic and sextic polynomial potentials with the Coulomb term and shown that the analytical results provide very good approximations to the numerical results.