# Quantization of the one-dimensional free harmonic oscillator as an example of the application of the node theorem and the MacDonald-Hylleraas-Undheim theorem

## TL;DR

This paper presents a simplified, intuitive approach to deriving the energy eigenvalues and eigenfunctions of the one-dimensional harmonic oscillator using heuristic arguments based on wavefunction properties.

## Contribution

It introduces a heuristic method leveraging the node theorem and MacDonald-Hylleraas-Undheim theorem for quantization, offering an alternative to traditional differential equation solutions.

## Key findings

- Energy eigenvalues derived heuristically
- Eigenfunctions characterized by wavefunction properties
- Simpler approach compared to traditional methods

## Abstract

Using heuristic arguments alone, based on the properties of the wavefunctions, we obtain the energy eigenvalues and the corresponding eigenfunctions of the one-dimensional harmonic oscillator. This approach is considerably simpler and is perhaps more intuitive than the traditional methods of solving a differential equation and manipulating operators.