Two electrons are placed into separate (very tiny) one-dimensional boxes $A$ and $B$. When the electrons are excited to higher energy levels and then decay to lower levels, they emit photons. The longest-wavelength spectral line from the electron in box $A$ is observed to have the same wavelength as the third-longest spectral line from the electron in box
$B$. The ratio of the box lengths $L_{A} / L_{B}$ is equal to
A) $3 / 7$
B) $7 / 3$
C) $\sqrt{3 / 7}$
D) $\sqrt{7 / 3}$
E) $9 / 49$

Question

Quizplus · Accepted Answer

The wavelength of the emitted photons is inversely proportional to the energy difference between the levels, which in turn is inversely proportional to the square of the box length. The longest wavelength corresponds to the transition between the first excited state and the ground state, while the third-longest wavelength in box B corresponds to a higher transition. The ratio of the lengths is thus related to the square root of the inverse ratio of the energy levels involved, leading to $\sqrt{3 / 7}$.

Two Electrons Are Placed into Separate (Very Tiny) One-Dimensional Boxes AAA

Two Electrons Are Placed into Separate (Very Tiny) One-Dimensional Boxes $A$