Strict Relationship: Potential - energy levels

TL;DR

This paper introduces a new analytical method to derive exact energy-potential relations in quantum systems, offering a simple, accurate, and computationally efficient alternative to numerical approaches for various potential models.

Contribution

The authors present a novel, non-numerical method for obtaining exact energy expressions as functions of potential, applicable to diverse potential forms with high accuracy.

Findings

Method accurately reproduces known energy levels for square well, harmonic oscillator, and Morse potentials.

The approach is simpler and more realistic than existing numerical methods.

Results show excellent agreement with previous analytical and numerical solutions.

Abstract

We have developed a new simple method to build the exact analytical expression of the eigenenergy as a function of the potential. The idea of our method is mainly based on the partitioning of the potential curve, solving the Schr\"odinger equation, realizing a discrete form of the energy quantification condition, and finally, deriving its integral form which permit to create a simple relation: Energy-potential. Our method has been applied to three examples: the well-known square well, the harmonic oscillators, and the Morse potential. Our non numerical method is more realistic, simpler, with high degree of accuracy, both satisfactory and not computationally complicated, and applicable for any forms of potential. Our results agree very well with the preceding ones.