General Quantization Rule

TL;DR

This paper introduces a general quantization rule for bound states in the Schrödinger equation, enabling straightforward calculation of energy levels for various systems, both exactly and non-exactly solvable.

Contribution

The paper presents a novel, unified quantization rule that simplifies energy level calculations across multiple quantum systems, regardless of their solvability.

Findings

Successfully applied to various 1D potentials including infinite well, harmonic oscillator, Morse, and Pöschl-Teller.

Extended to 3D systems like the hydrogen atom and harmonic oscillator.

Provides a general method for calculating bound state energies efficiently.

Abstract

A general quantization rule for bound states of the Schrodinger equation is presented. Like fundamental theory of integral, our idea is mainly based on dividing the potential into many pieces, solving the Schr\"odinger equation, and deriving the general quantization rule. For both exactly and non-exactly solvable systems, the energy levels of all the bound states can be easily calculated from the general quantization rule. Using this new general quantization rule, we re-calculate the energy levels for the one-dimensional system, with an infinite square well, with the harmonic oscillator potential, with the Morse Potential, with the symmetric and asymmetric Rosen-Morse potentials, with the first P\"oschl-Teller potential, with the Coulomb Potential, with the V-shape Potential, and the ax^4 potential, and for the three dimensions systems, with the harmonic oscillator potential, with the ordinary Coulomb potential, and for the hydrogen atom.