The electron in a hydrogen atom makes a transition n_1 → n_2 where n_1 and n_2 are the principal quantum numbers of the two states. Assume the Bohr model to be valid….

Q: The electron in a hydrogen atom makes a transition n₁ → n₂ where n₁ and n₂ are the principal quantum numbers of the two states. Assume the Bohr model to be valid. The time period of the electron in the
initial state is eight times that in the final state. What are the possible values of n₁ and n₂ ?

Sol: $\large T \propto n^3 $

$\large \frac{T_1}{T_2} = \frac{n_1^3}{n_2^3} $

As , T₁ = 8 T₂

$\large 8 = \frac{n_1^3}{n_2^3} $

n₁ = 2 n₂

Thus the possible values of n₁ and n₂ are n₁ = 2 , n₂ = 1. n₁ = 4, n₂ = 2 , n₁ = 6 , n₂ = 3; and so on.