Question

An excited state of an atom has a $25ns$lifetime. What is the probability that an excited atom will emit a photon during a $0.50ns$ interval?

Short answer

The probability that an excited atom will emit a photon during a $0.50ns$interval is $0.20\%$

Step-by-step solution

Given Information 

We need to find the probability that an excited atom will emit a photon during a $0.50ns$interval.

Simplify

Consider :

$$\begin{gathered} \tau =25ns \\ ∆t=0.50ns \end{gathered}$$

Using formula:

$∆t=r∆t$.

First, finding the decay rate:

$$\begin{gathered} r=\frac{1}{\tau } \\ r=\frac{1}{25ns} \\ r=0.040n{s}^{-1} \end{gathered}$$

Now, find prob decay $∆t$:

$$\begin{gathered} ∆t=r\times ∆t \\ ∆t=\left(0.040n{s}^{-1}\right)\left(0.50ns\right) \\ ∆t=0.02\approx 0.2\% \end{gathered}$$