Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 39 - Wave Functions and Uncertainty - Exercises and Problems - Page 1174: 6

Answer

${\bf 18\;000}\;\rm photon$

Work Step by Step

We know that the probability of finding a photon within a narrow region of width $\delta x$ at position $x$ is given by $$\text{Prob}( \text{in } \delta x \text{ at } x) \propto |A(x)|^2 \delta x$$ This means that the probability of finding a photon in a small region $\delta x $ at position $x $ is proportional to the square of the amplitude of the light wave $ |A(x)|^2 $ times the width of the region $\delta x $. So the ratio of probabilities at two different positions $x_1$ and $x_2$: $$ \frac{\text{Prob(in } \delta x_1 \text{ at } x_1)}{\text{Prob(in } \delta x_2 \text{ at } x_2)} = \frac{|A(x_1)|^2 \delta x}{|A(x_2)|^2 \delta x} $$ Since the width $ \delta x $ is the same, they cancel out, $$ \frac{\text{Prob(in } \delta x_1 \text{ at } x_1)}{\text{Prob(in } \delta x_2 \text{ at } x_2)} = \frac{|A(x_1)|^2}{|A(x_2)|^2} $$ Hence, $$\dfrac{\dfrac{N_1}{N_{tot}}}{\dfrac{N_2}{N_{tot}}} = \frac{|A(x_1)|^2}{|A(x_2)|^2} $$ where $N_{tot}$ is the total number of photons, $ N_1$ is the number of photons detected at position $ x_1 $ in the region $\delta x$, and $ N_2 $ is the number of photons detected at position $ x_2 $ in the region $\delta x$ $N_{tot}$ they cancel out, $$\dfrac{N_1}{ N_2} = \frac{|A(x_1)|^2}{|A(x_2)|^2}$$ Thus, $$N_2=\dfrac{N_1|A(x_2)|^2}{|A(x_1)|^2}$$ Plug the known; $$N_2=\dfrac{2000 (30)^2}{(10)^2}$$ $$N_2=\color{red}{\bf 18\;000}\;\rm photon$$
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