## Conceptual Physics (12th Edition)

Energy is conserved. The sum of the two photon energies equals the single photon's energy. $$E_{1} + E_{2} = E_{3}$$ For photons, E = hf, where E is the energy, h is Planck's constant, and f is the photon's vibrational frequency. Wavelength is inversely proportional to frequency. That is, the speed of light is frequency multiplied by wavelength. $$c = f \lambda$$ Solve for $\lambda = \frac{c}{f}$. Finally, we are ready to relate the 3 wavelengths. $$E_{1} + E_{2} = E_{3}$$ $$h(\frac{c}{\lambda_{1}} +\frac{c}{\lambda_{2}}) = h \frac{c}{\lambda_{3}}$$ $$\frac{1}{\lambda_{1}} +\frac{1}{\lambda_{2}} = \frac{1}{\lambda_{3}}$$ So the sum of the reciprocals of the two photon wavelengths, emitted when the electron transitions from 4 to 3 and then from 3 to 1, is equal to the reciprocal of the wavelength of light emitted when the electron transitions from quantum level 4 straight down to quantum level 1.