The Bounded Anharmonic Oscillators: a simple approach

TL;DR

This paper introduces a straightforward, accurate, and versatile method for solving anharmonic oscillator equations by partitioning the potential and applying continuity conditions, applicable to various potential forms.

Contribution

The authors present a novel, simple approach to solve anharmonic oscillators that is more realistic, accurate, and computationally efficient than existing methods.

Findings

Method yields results in excellent agreement with previous studies.

Applicable to diverse potential types including Morse and bounded oscillators.

Provides exact wave functions for energy levels.

Abstract

We have developed a simple method to solve anharmonic oscillators equations. The idea of our method is mainly based on the partitioning of the potential curve into (n+1) small intervals, solving the Schr\"odinger equation in each subintervals, and writing the continuity conditions of each solutions at each subintervals boundary, which leads to the energy quantification condition, so to the energy levels, and finally, one can calculate the exact wave function associated to each energy level. Our method has been applied to three examples: the well-known square well, the bounded and unbounded harmonic oscillators, the Morse potential, and some anharmonic oscillators bounded by two infinite walls. Our 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.