What is the ratio of wavelength λ_1/λ_2 in the given figure of energy…

The correct answer is Option (4) → \( 6 : 1 \)

When a system drops from a higher level to a lower one, it emits a photon whose energy equals the energy difference of the two levels, \( \Delta E = hc/\lambda \). So the wavelength is inversely proportional to the energy difference.

Reading the levels from the figure: \( \lambda_1 \) is emitted in the jump from \( -E \) to \( -5E/3 \), and \( \lambda_2 \) in the jump from \( -E \) to \( -5E \).

    \( \displaystyle \Delta E_1 = -E - \left(-\frac{5}{3}E\right) = \frac{2}{3}E \)

    \( \displaystyle \Delta E_2 = -E - (-5E) = 4E \)

    \( \displaystyle \frac{\lambda_1}{\lambda_2} = \frac{\Delta E_2}{\Delta E_1} = \frac{4E}{2E/3} = 6 \)

So \( \lambda_1 : \lambda_2 = 6 : 1 \). The smaller energy jump gives the longer wavelength.