Q. The electron in a hydrogen atom makes a transition $n_{1} \rightarrow n_{2}$where $n_{1}$and $n_{2}$ are the principal quantum numbers of the two states. Assume the Bohr model to be valid. The time period of electron in the initial state is 8 times that in the final state. The possible values of $n_{1}\ and\ n_{2}$ are
A) $n_{1} = 6,n_{2} = 3$
B) $n_{1} = 8,\ n_{2} = 2$
C) $n_{1} = n_{2} = 1$
D) $n_{1} = 8,\ n_{2} = 1$
Explanation
(a) As $T = \frac{2\pi r}{v}$ or $V = \frac{nh}{2\pi mr}$
$\therefore\ T = \frac{2\pi r}{nh/2\pi mr} = \frac{mr^{2}}{nh} \propto \frac{r^{2}}{n}$
But $r \propto n^{2}\therefore T \propto n^{3}$
or $\frac{T_{1}}{T_{2}} = \left( \frac{n_{1}}{n_{2}} \right)^{3}$,As $T_{1} = 8T_{2}$
$\therefore\ \left( \frac{n_{1}}{n_{2}} \right)^{3} = 8,\frac{n_{1}}{n_{2}} = 2$
Therefore, in given values $n_{1} = 6,\ n_{2} = 3$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.