JEE Main · 2017 · 02 Apr 2017 · Offline · Modern Physics

Since $E=\frac{hc}{\lambda }$, each transition's wavelength is inversely proportional to its energy gap, i.e. $\lambda \propto \frac{1}{\Delta E}$. Reading the level diagram, ${\lambda }_1$ is the jump from $-E$ down to $-2E$, an energy gap of $\Delta E_1=E$, while ${\lambda }_2$ is the jump from $-E$ down to $-\frac{4}{3}E$, a gap of $\Delta E_2=\frac{E}{3}$. Hence $r=\frac{{\lambda }_1}{{\lambda }_2}=\frac{\Delta E_2}{\Delta E_1}=\frac{E/3}{E}=\frac{1}{3}$.