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

Abstract

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

Keywords: Time-independent Schrödinger equation, MacDonald-Hylleraas-Undheim theorem, Node theorem, Hermite polynomials,  energy eigenvalues