Quantum mechanics is about probabilities. We cannot say exactly where the electron is located, but we can use quantum mechanics to calculate the probability that the electron is located in a small interval $\delta x$ at position $x$. Similarly, we cannot say exactly when an excited electron will undergo a quantum jump and emit a photon. However, we can use quantum mechanics to find the probability that the electron will undergo a quantum jump during a small time interval $\delta t$ at time $t$.

Let us assume that the probability of an excited atom emitting a photon during time interval $\delta t$ is independent of how long the atom has been waiting in the excited state. For example, a newly excited atom may have a 10% probability of emitting a photon within the 1 ns interval from 0 ns to 1 ns. If it survives until $t=7\text{ns,}$ our assumption is that it still has a 10% probability of emitting a photon during the 1 ns interval from 7 ns to 8 ns.

This assumption, which can be justified with a detailed analysis, is similar to flipping coins. The probability of a head on your first flip is 50%. If you flip seven heads in a row, the probability of a head on your eighth flip is still 50%. It is unlikely that you will flip seven heads in a row, but doing so does not influence the eighth flip. Likewise, it may be unlikely for an excited atom to live for 7 ns, but doing so does not affect its probability of emitting a photon during the next 1 ns.

If $\delta t$ is small, the probability of photon emission during time interval $\delta t$ is directly proportional to $\delta t$. That is, if the emission probability in 1 ns is 1%, it will be 2% in 2 ns and 0.5% in 0.5 ns. (This logic fails if $\delta t$ gets too big. If the probability is 70% in 20 ns, we can not say that the probability would be 140% in 40 ns because a probability $>1$ is meaningless.) We will be interested in the limit $\delta t\to dt$, so the concept is valid and we can write

where $r$ is called the decay rate because the number of excited atoms decays with time. It is a probability per second, with units of $\text{s}{}^{-1}$, and thus is a rate. For example, if an atom has a 5% probability of emitting a photon during a 2 ns interval, its decay rate is

FIGURE 41.29 shows $N_{\text{exc}}$ atoms in an excited state. During a small time interval $\delta t$, the number of these atoms that we expect to undergo a quantum jump and emit a photon is $N_{\text{exc}}$ multiplied by the probability of decay. That is,

FIGURE 41.29 The number of atoms that emit photons during $\text{Δ}t$ is directly proportional to the number of excited atoms.

Now the change in $N_{\text{exc}}$ is the negative of Equation 41.20. For example, suppose 1000 excited atoms are present at time t and each has a 5% probability of emitting a photon in the next 1 ns. On average, the number of photons emitted during the next 1 ns will be $1000\times 0.05=50$. Consequently, the number of excited atoms changes by $\text{Δ}{N}_{\text{exc}}=-50$, with the minus sign indicating a decrease.

Thus the change in the number of atoms in the excited state is

Now let $\delta t\to dt$. Then $\text{Δ}{N}_{\text{exc}}\to d{N}_{\text{exc}}$ and Equation 41.21 becomes

Equation 41.22 is a rate equation because it describes the rate at which the excited-state population changes. If $r$ is large, the population will decay at a rapid rate and will have a short lifetime. Conversely, a small value of $r$ implies that the population will decay slowly and will live a long time.

The rate equation is a differential equation, but we solved a similar equation for RC circuits in Chapter 28. First, we rewrite Equation 41.22 as

Then we integrate both sides from $t=0$, when the initial excited-state population is $N_0$, to an arbitrary time t when the population is $N_{\text{exc}}$. That is,

Both are well-known integrals, giving

We can solve for the number of excited atoms at time t by taking the exponential of both sides, then multiplying by $N_0$. Doing so gives

Notice that $N_{\text{exc}}=N_0$ at $t=0$, as expected. Equation 41.24, the decay equation, shows that the excited-state population decays exponentially with time, as we saw in the experimental data of Figure 41.28.

It will be more convenient to write Equation 41.24 as

where

This is the definition of the lifetime we used in Equation 41.18 to describe the experimental results. The lifetime is the inverse of the decay rate $r$.