Question

FIGURE Q 39.1 shows the probability density for photons to be detected on the x-axis.

a. Is a photon more likely to be detected at x = 0 m or at x=1 m? Explain.

b. One million photons are detected. What is the expected number of photons in a 1 mm-wide interval at x = 0.50 m?

Short answer

The expected number of photons is: $1,000$

Step-by-step solution

Given information

FIGURE Q 39.1 shows the probability density for photons to be detected on the x-axis.

Explanation

a) The photon is detected most likely at $x\approx 1$, because the probability density is higher there.

Explanation

$\text{b)}\mathrm{P}\left(\text{in}\delta x\text{at}x\right)=\mathrm{P}\left(x\right)\delta x\text{.}$

If 1,000,000 photons are detected then the number expected in al-mm-wide interval is 1,000,000$$\begin{gathered} \mathrm{P}\left(\mathit{x}\right)\delta \mathit{x}\text{where where}\delta x=1.0\mathrm{mm}.Forx=0.50\mathrm{m},\mathrm{P}\left(x\right)=1m^{-1} \\ \text{and the expected number is}(1,000,000)\left(1m^{-1}\right)(0.001m)=1,000\text{.} \end{gathered}$$