$$\hat{H}=\frac{\hat{p}^2}{2m}+\frac{m\omega^2\hat{x}^2}{2}$$

$$E\psi(x)=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi(x)+\frac{m\omega^2x^2}{2}\psi(x)$$

Substitution \(x=\frac{y}{\sqrt{m\omega/\hbar}}\), \(\phi(y)=\psi\left(\frac{y}{\sqrt{m\omega/\hbar}}\right)\)
$$-\frac{d^2}{dy^2}\phi(y)+y^2\phi(y)=\frac{2E}{\hbar\omega}\phi(y)$$
$$\phi(y)=H(y)\exp\left(-\frac{y^2}{2}\right)$$
Hermite's differential equation
$$-\frac{d^2}{dy^2}H(y)+2y\frac{d }{dy}H(y)+\left(1-\frac{2E}{\hbar\omega}\right) H(y)=0$$
A square integrable solution
$$H(y)=\sum_{j=0}a_j y^j,\quad a_{j+2}=\frac{2j+1-\frac{2E}{\hbar\omega}}{(j+2)(j+1)}a_j$$
The energy spectrum
$$E_n=\hbar\omega\left(n+\frac{1}{2}\right),\quad n=0,1,...$$
Eigenvectors
$$\psi_n(x)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}H\left(x\sqrt{\frac{m\omega}{\hbar}}\right)\frac{\exp\left(-\frac{m\omega}{2\hbar}x^2\right)}{2^{n/2}\sqrt{n!}}$$
Hermite's polynomials
$$H_n(y)=(-1)^n e^{y^2}\frac{d^n}{dy^n}e^{-y^2}$$