Highly Accurate Analytic Presentation of Solution of the Schr\"{o}dinger Equation

TL;DR

This paper introduces a novel analytic method to approximate solutions of the Schrödinger equation for various potentials, providing high-precision energy and wave function estimates useful for physical system analysis.

Contribution

It develops an analytic approach using the quasi-linearization method to solve the Schrödinger equation in a new way, applicable to arbitrary potentials.

Findings

Accurate analytic expressions for energies and wave functions were derived.

The method was demonstrated on the Yukawa potential.

Results enable precise parameter effect estimations.

Abstract

High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schr\"{o}dinger equation is cast into nonlinear Riccati equation, which is solved analytically in first iteration of the quasi-linearization method (QLM). The zeroth iteration is based on general features of the exact solution near the boundaries. The approach is illustrated on the Yukawa potential. The results enable accurate analytical estimates of effects of parameter variations on physical systems.