A new non-perturbative approach in quantum mechanics for time-independent Schr\"{o}dinger equations

TL;DR

This paper introduces a non-perturbative method based on the homotopy analysis method (HAM) to solve time-independent Schrödinger equations, enabling accurate solutions without reliance on small parameters, applicable to quantum mechanics and QCD.

Contribution

It presents a novel HAM-based approach for solving nonlinear Schrödinger equations that remains valid even with large disturbances, surpassing traditional perturbative methods.

Findings

Successfully applied to nonlinear harmonic oscillator

Provides convergent solutions for large disturbances

Potential for improved experimental data validation

Abstract

A new non-perturbative approach is proposed to solve time-independent Schr\"{o}dinger equations in quantum mechanics and chromodynamics (QCD). It is based on the homotopy analysis method (HAM), which was developed by the author for highly nonlinear equations since 1992 and has been widely applied in many fields. Unlike perturbative methods, this HAM-based approach has nothing to do with small/large physical parameters. Besides, convergent series solution can be obtained even if the disturbance is far from the known status. A nonlinear harmonic oscillator is used as an example to illustrate the validity of this approach for disturbances that might be more than hundreds larger than the possible superior limit of the perturbative approach. This HAM-based approach could provide us rigorous theoretical results in quantum mechanics and chromodynamics (QCD), which can be directly compared with experimental data. Obviously, this is of great benefit not only for improving the accuracy of experimental measurements but also for validating physical theories.