The electron in a hydrogen atom jumps back from an excited state to ground state, by emitting a phot

Question

The electron in a hydrogen atom jumps back from an excited state to ground state, by emitting a photon of wavelength $\lambda_0=\frac{16}{15 R}$, where $R$ is Rydberg's constant. In place of emitting one photon, the electron could come back to ground state by

Professor PC Thomas Classes · Answer

(a): Using the Rydberg's Formula for the given transition \ $ \frac{1}{\lambda_0}=R\left[\frac{1}{1^2}-\frac{1}{n^2}ight] $ \ Solving we get $n=4$. Thus electron can come to ground state by emitting two or three photons such that the sum of energies of each transition remain same so we can use $ \ \begin{aligned} & \frac{h c}{\lambda_0}=\frac{h c}{\lambda_1}+\frac{h c}{\lambda_2} \Rightarrow \frac{1}{\lambda_0}=\frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\frac{15 R}{16} \ & 	ext { and } \frac{h c}{\lambda_0}=\frac{h c}{\lambda_1}+\frac{h c}{\lambda_2}+\frac{h c}{\lambda_3} \Rightarrow \frac{1}{\lambda_0}=\frac{1}{\lambda_1}+\frac{1}{\lambda_2}+\frac{1}{\lambda_3}=\frac{15 R}{16} \end{aligned} $