Exponential Decay

We have computed transition rates using our theory of radiation. In doing this, we have assumed that our calculations need only be valid near $t=0$. More specifically, we have assumed that we start out in some initial state $i$ and that the amplitude to be in that initial state is one. The probability to be in the initial state will become depleted for times on the order of the lifetime of the state. We can account for this in terms of the probability to remain in the initial state.

Assume we have computed the total transition rate.

$$\bgroup\color{black} \Gamma_{tot}=\sum\limits_n\Gamma_{i\rightarrow n} \egroup$$

This transition rate is the probability per unit time to make a transition away from the initial state evaluated at $t=0$. Writing this as an equation we have.

$$\bgroup\color{black} \left.{dP_i\over dt}\right\vert _{t=0}=-\Gamma_{tot} \egroup$$

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.

$$\bgroup\color{black} {dP_i(t)\over dt}=-\Gamma_{tot}P_i(t) \egroup$$

The solution to this simple first order differential equation is

$$\bgroup\color{black} P_i(t)=P_i(t=0)e^{-\Gamma_{tot}t} \egroup$$

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 $t=0$. Clearly the number of atoms remaining in the 2p state will obey the equation

$$\bgroup\color{black} {dN_{2p}(t)\over dt}=-\Gamma_{tot}N_{2p}(t) \egroup$$

and we will have our exponential time distribution.

We may define the lifetime of a state to the the time after which only ${1\over e}$ of the decaying state remains.

$$\bgroup\color{black} \tau = {1\over\Gamma_{tot}} \egroup$$