Suppose the energy of the longest-wavelength photon emitted by an electron in a box of length $L$ is $E_{\mathrm{ph}}$. What is the energy of the photon with the fourth-longest wavelength that might be emitted?
A) $(4 / 3) E_{\mathrm{ph}}$
B) $(5 / 3) E_{\mathrm{ph}}$
C) $(7 / 3) E_{\mathrm{ph}}$
D) $(8 / 3) E_{\text {ph }}$
E) $3 E_{\text {ph }}$

Question

Quizplus · Accepted Answer

The energy levels of an electron in a box are quantized and proportional to the square of the quantum number $n$. The energy difference between levels, which corresponds to the energy of the emitted photon, increases with $n$. The longest wavelength (lowest energy) photon corresponds to a transition from $n=2$ to $n=1$, which has an energy proportional to $3E_{\mathrm{ph}}$ (since $E \propto n^2$ and the energy difference for the first transition is $2^2 - 1^2 = 3$). The fourth-longest wavelength corresponds to a transition from $n=5$ to $n=4$, with an energy proportional to $5^2 - 4^2 = 25 - 16 = 9$, which is $3$ times the energy of the longest wavelength photon. Therefore, the energy of the photon with the fourth-longest wavelength is $(8 / 3) E_{\mathrm{ph}}$, considering the initial reference energy level difference was $3E_{\mathrm{ph}}$.

Suppose the Energy of the Longest-Wavelength Photon Emitted by an Electron