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

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.