(i) Statement of Bohr's quantization condition
de- Broglie explanation of stationary orbits
(ii) Relation between ${\mathit{\lambda }}_1,{\mathit{\lambda }}_2{\mathit{\lambda }}_3$
(i) Only those orbits are stable for which the angular momentum, of revolving electron, is an integral multiple of $\frac{\mathit{h}}{2\mathit{\pi }}$ .
[Alternatively
$\mathit{L}=\frac{\mathit{n}\mathit{h}}{2\mathit{\pi }}$ i.e. angular momentum of orbiting electron is quantized.]
According to de-Broglie hypothesis
Linear momentum $\left(\mathit{p}\right)=\frac{\mathit{h}}{\mathit{\lambda }}$
And for circular orbit $\mathit{L}={\mathit{r}}_{\mathit{n}}\mathit{p}$ where ' $\mathit{r}$ ' is the radius of quantized orbits.
$=\frac{\mathit{r}\mathit{h}}{\mathit{\lambda }}$Also $\mathit{L}=\frac{\mathit{n}\mathit{h}}{2\mathit{\pi }}$
$\therefore \frac{\mathit{r}\mathit{h}}{\mathit{\lambda }}=\frac{\mathit{n}\mathit{h}}{2\mathit{\pi }}$
$\Rightarrow 2\mathit{\pi }{\mathit{r}}_{\mathit{n}}=\mathit{n}\mathit{\lambda }$
$\therefore$ Circumference of permitted orbits are integral multiples of the wave-length $\mathit{\lambda }$.
(ii) ${\mathit{E}}_{\mathit{C}}-{\mathit{E}}_{\mathit{B}}=\frac{\mathit{h}\mathit{c}}{{\mathit{\lambda }}_1}$ .......(i)
${\mathit{E}}_{\mathit{B}}-{\mathit{E}}_{\mathit{A}}=\frac{\mathit{h}\mathit{c}}{{\mathit{\lambda }}_2}$ .......(ii)
${\mathit{E}}_{\mathit{C}}-{\mathit{E}}_{\mathit{A}}=\frac{\mathit{h}\mathit{c}}{{\mathit{\lambda }}_3}$ .......(iii)
Adding (i) & (ii)
${\mathit{E}}_{\mathit{C}}-{\mathit{E}}_{\mathit{A}}=\frac{\mathit{h}\mathit{c}}{{\mathit{\lambda }}_1}+\frac{\mathit{h}\mathit{c}}{{\mathit{\lambda }}_2}$ ........(iv)
Using equation (iii) and (iv)
$\frac{\mathit{h}\mathit{c}}{{\mathit{\lambda }}_3}=\frac{\mathit{h}\mathit{c}}{{\mathit{\lambda }}_1}+\frac{\mathit{h}\mathit{c}}{{\mathit{\lambda }}_2}$
$\Rightarrow \frac{1}{{\mathit{\lambda }}_3}=\frac{1}{{\mathit{\lambda }}_1}+\frac{1}{{\mathit{\lambda }}_2}$