Quantum Harmonic Oscillator

incomplete The model for a classical harmonic oscillator is given by:

$$v=\frac{1}{2}{k}_0{x}^2$$

Where $k_0$ is the spring constant

$$F=-\frac{dv}{dx}=-k_{0}x=m\frac{d{x}^2}{d{t}^2}$$

This differential equation yields solutions:

$$e^{i\omega t}and\omega \equiv \sqrt{\frac{k_0}{m}}$$ $$H=\frac{p^2}{2m}+V(x)$$

Where H represents the energy of the system.

If we extend this to the quantum case, we get

$$\hat{H}=\frac{{\hat{p}}^2}{2m}+v(\hat{x})⟹=\frac{{\hslash }^2}{2m}\frac{d^2\phi (x)}{d{x}^2}+v(x)\phi (x)=E\phi (x)$$