The Photon, the Quantum of Light — Question 5

Question 5

Total power and photon rate reveal an unknown two-color mixture

A beam contains photons of energies 2.00 eV and 3.00 eV only. Its total power is $4.00\,\mathrm{mW}$ and total photon rate is $10^{16}\,\mathrm{s^{-1}}$. Use $1\,\mathrm{eV}=1.60\times10^{-19}\,\mathrm J$.

1. Find the fraction of photons in each energy group.

2. Find the optical power carried by each group and check both conservation constraints.

Question 5 - Solution

Total power and photon rate reveal an unknown two-color mixture

(a) Method and calculation

$$\overline E=P/\dot N=2.50\,\mathrm{eV},\quad2f+3(1-f)=2.50.$$$$\boxed{f=1/2},\qquad\dot N_2=\dot N_3=5.00\times10^{15}\,\mathrm{s^{-1}}.$$

(b) Interpretation and further calculation

$$\boxed{P_2=1.60\,\mathrm{mW}},\qquad\boxed{P_3=2.40\,\mathrm{mW}}.$$$$P_2+P_3=4.00\,\mathrm{mW},\quad\dot N_2+\dot N_3=10^{16}\,\mathrm{s^{-1}}.$$

Physical check

The inferred mean energy lies between the two component energies; otherwise the proposed two-color model would be impossible.