# Why the excited states of an atom have an energy width?

All my experience with textbook problems of quantum mechanics shows that the energy levels associated with the bound states of a confined quantum system are discrete and sharp. For example, the energy levels of the hydrogen atom. Why is it then said that excited states of an atom have an energy width? Where does the width come from? This fact doesn't match with the examples I know in quantum mechanics. If this question is asked before can some one give the links?

$$\Gamma_{rad}(\omega) = \frac{4\, \alpha\, \omega^3\,| \langle 1|\mathbf{r}|2\rangle |^2}{3 \,c^2}$$
with $\alpha$ being the fine structure constant, $\omega$ the center frequency and $\langle 1|\mathbf{r}|2\rangle$ being the overlap between the two electronic states on either side of the transition.