Calculate the Average Lifetime of an Excited Atomic State

We can use the formula for the average lifetime of an excited state:

$\tau = \frac{\hbar}{\Gamma}$

where $\hbar = h/2\pi$ is the reduced Planck constant and $\Gamma$ is the linewidth of the excited state, related to its lifetime by:

$\Gamma = \frac{\hbar}{\tau}$

The energy uncertainty of the excited state corresponds to a linewidth by the relation:

$\Delta E \Delta t = \frac{\hbar}{2}$

We can use the given energy uncertainty to find the linewidth:

$\Delta E = 0.3\text{ eV} = 0.3\times 1.602\times 10^{-19}\text{ J} = 4.806\times 10^{-20}\text{ J}$

$\Delta t = \frac{\hbar}{2\Delta E} = \frac{6.626\times 10^{-34}\text{ J.s}}{2\times 4.806\times 10^{-20}\text{ J}} = 6.906\times 10^{-15}\text{ s} = 6.906\text{ fs}$

The linewidth is then:

$\Gamma = \frac{\hbar}{\Delta t} = \frac{6.626\times 10^{-34}\text{ J.s}}{6.906\times 10^{-15}\text{ s}} = 9.582\times 10^{-20}\text{ J}$

The average lifetime is then:

$\tau = \frac{\hbar}{\Gamma} = \frac{6.626\times 10^{-34}\text{ J.s}}{9.582\times 10^{-20}\text{ J}} = 6.918\times 10^{-15}\text{ s} = \boxed{6.918\text{ fs}}$

rounded to three decimal places.