Harmonikus oszcillátor energiája, Hamilton operátora

$E_n = \hbar \omega (n + \frac{1}{2}), \quad n=0,\,1,\,2,\, \ldots$

$\mathcal{H} = \hbar \omega \left(\hat{N} + \frac{1}{2}\right) = \hbar \omega \left(\hat{a}^{\dagger}\hat{a} + \frac{1}{2}\right)= \frac{\hat{p}^2}{2m} + \frac{m \omega^2}{2}\hat{x}^2.$