Assume we have computed the total transition rate.

This transition rate is the probability per unit time to make a transition away from the initial state evaluated at . Writing this as an equation we have.

For larger times we can assume that the probability to make a transition away from the initial state is proportional to the probability to be in the initial state.

The solution to this simple first order differential equation is

If you are having any trouble buying this calculation, think of a large ensemble of hydrogen atoms prepared to be in the 2p state at . Clearly the number of atoms remaining in the 2p state will obey the equation

and we will have our exponential time distribution.

We may define the lifetime of a state to the the time after which only
of the decaying state remains.

Jim Branson 2013-04-22