TimeEnergy Uncertainty Problem

[Leonhard]

1. The problem statement, all variables and given/known data

In laser physics a common approximation for the time-energy uncertainty is given by.

\Delta \omega \cdot \tau \geq \approx 2 \pi

The problem is to use the Energy-Time Uncertainty relationship to derive a more exact answer than this.

2. Relevant equations

\sqrt{\left\langle E^2 \right\rangle \left\langle t^2 \right\rangle} \geq \frac{\hbar}{2}

It should be noted that we are supposed to use FWHM for the uncertainties, so the following relationship is supplied.

\Delta_{FWHM} x = 2\sqrt{2 ln(2)}\sqrt{\left\langle x^2 \right\rangle}

3. The attempt at a solution

\sqrt{\left\langle E^2 \right\rangle \left\langle t^2 \right\rangle} = \frac{\hbar}{2}

I substitute

E = \hbar \omega

Giving

\sqrt{\left\langle \hbar^2 \omega^2 \right\rangle \left\langle t^2 \right\rangle} = \frac{\hbar}{2}

I divide with \hbar on both sides

\sqrt{\left\langle \omega^2 \right\rangle \left\langle t^2 \right\rangle} = \frac{1}{2}

We then substitute

\tau = 2\sqrt{2 ln(2)} \sqrt{\left\langle t^2 \right\rangle}

And

\Delta \omega = 2\sqrt{2 ln(2)} \sqrt{\left\langle \omega^2 \right\rangle}

Giving

\Delta \omega \cdot \tau = 4 ln(2)

I don't know where I've done anything wrong, but I would appreciate some help a lot :3