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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.

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