# TimeEnergy Uncertainty Problem

1. Nov 12, 2009

### 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