Time Energy Uncertainty

Time is not an observable.

We could count $N$ full waves of frequency $\omega$ over a time window $T$ . You can only count to within $\pm 1$ wave, so $\Delta N\geq 1$ .

The uncertainty in the wavecount is $\Delta N$

We know that

N=\frac{\omega T}{2\pi} \quad\Rightarrow\quad \Delta N = \frac{\Delta \omega T}{2\pi} \geq 1 \quad \Rightarrow\quad \Delta \omega T \geq 2\pi

From (Planck-Einstein relation),

\Delta E \cdot T\geq h = \frac{\hbar}{2}