Conceptual Physics (12th Edition)

Published by Addison-Wesley
ISBN 10: 0321909100
ISBN 13: 978-0-32190-910-7

Chapter 30 - Think and Explain: 80

Answer

Yes, there is a relationship among the wavelengths.

Work Step by Step

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.
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