Physics — Quantum Harmonic Oscillator

E_n = \hbar\omega\left(n + \tfrac{1}{2}\right)

\hat{H} = \frac{\hat{p}^2}{2m} + \tfrac{1}{2}m\omega^2 \hat{x}^2

\hat{a},\ \hat{a}^\dagger\ \text{(ladder operators)}

E_0 = \tfrac{1}{2}\hbar\omega\ \text{(zero-point)}