Elementary quantization of the Harmonic Oscillator in one and three dimensions in coordinate representation

The one dimensional case can be extended easily to the three dimensional case; one might not be satisfied with such a solution. We will also demonstrate the solution in spherical coordinates. With a harmonic oscillator potential, Schrödinger's equation in one dimension becomes,

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}m\omega^2 x^... ...= \frac{2E}{\hbar \omega} \rightarrow \ \frac{d^2\psi}{d\eta^2}=(\eta^2 -K)\psi$$ (2.1)

A solution to this equation can be found with the traditional approach.